It is widely accepted that physical exercise can be used as a tool for the prevention and treatment of various diseases or disorders. In addition, in the recent years, exercise has also been successfully used to enhance people's cognition. There is a large amount of research that has supported the benefits of physical exercise on human cognition, both in children and adults. Among these studies, some have focused on the acute or transitory effects of exercise on cognition, while others have focused on the effects of regular physical exercise. However, the relation between exercise and cognition is complex and we still have limited knowledge about the moderators and mechanisms underlying this relation. Most of human studies have focused on the behavioral aspects of exercise-effects on cognition, while animal studies have deepened in its possible neuro-physiological mechanisms. Even so, thanks to advances in neuroimaging techniques, there is a growing body of evidence that provides valuable information regarding these mechanisms in the human population. This review aims to analyze the effects of regular and acute aerobic exercise on cognition. The exercise-cognition relationship will be reviewed both from the behavioral perspective and from the neurophysiological mechanisms. The effects of exercise on animals, adult humans, and infant humans will be analyzed separately. Finally, physical exercise intervention programs aiming to increase cognitive performance in scholar and workplace environments will be reviewed.
Citation: Blai Ferrer-Uris, Maria Angeles Ramos, Albert Busquets, Rosa Angulo-Barroso. Can exercise shape your brain? A review of aerobic exercise effects on cognitive function and neuro-physiological underpinning mechanisms[J]. AIMS Neuroscience, 2022, 9(2): 150-174. doi: 10.3934/Neuroscience.2022009
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It is widely accepted that physical exercise can be used as a tool for the prevention and treatment of various diseases or disorders. In addition, in the recent years, exercise has also been successfully used to enhance people's cognition. There is a large amount of research that has supported the benefits of physical exercise on human cognition, both in children and adults. Among these studies, some have focused on the acute or transitory effects of exercise on cognition, while others have focused on the effects of regular physical exercise. However, the relation between exercise and cognition is complex and we still have limited knowledge about the moderators and mechanisms underlying this relation. Most of human studies have focused on the behavioral aspects of exercise-effects on cognition, while animal studies have deepened in its possible neuro-physiological mechanisms. Even so, thanks to advances in neuroimaging techniques, there is a growing body of evidence that provides valuable information regarding these mechanisms in the human population. This review aims to analyze the effects of regular and acute aerobic exercise on cognition. The exercise-cognition relationship will be reviewed both from the behavioral perspective and from the neurophysiological mechanisms. The effects of exercise on animals, adult humans, and infant humans will be analyzed separately. Finally, physical exercise intervention programs aiming to increase cognitive performance in scholar and workplace environments will be reviewed.
attention deficit hyperactivity disorder;
brain-derived neurotrophic factor;
insulin-like growth factor 1
In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see [1,2].
Let S:K→R+ be a convex function on a convex set K and ρ,ς∈K with ρ≠ς. Then,
S(ρ+ς2)≤1ς−ρ∫ςρS(ϖ)dϖ≤S(ρ)+S(ς)2. | (1) |
In [3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).
Let S:K→R+ be a convex function on a convex set K and ρ,ς ∈K with ρ≠ς. Then,
S(ρ+ς2)≤1∫ςρD(ϖ)dϖ∫ςρS(ϖ)D(ϖ)dϖ≤S(ρ)+S(ς)2∫ςρD(ϖ)dϖ. | (2) |
If D(ϖ)=1, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya [4] proved the Hadamard type inequality for coordinated convex functions such that
Let G:Δ→R+ be a coordinate convex function on Δ=[ς,ρ]×[μ,ν]. If G is double fractional integrable, then following inequalities hold:
G(μ+ν2,ς+ρ2)≤Γ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)]+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)]≤Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)]≤Γ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+Iαν−G(μ,ς)+Iαν−G(μ,ρ)]+Γ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)˜+Iβρ−G(ν,ς)+Iβς+G(μ,ρ)+Iβρ−G(ν,ς)]≤G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (3) |
If α=1, then we obtain the following Dragomir inequality [5] on coordinates:
G(μ+ν2,ς+ρ2) |
≤12[1ν−μ∫νμG(x,ς+ρ2)dx+1ρ−ς∫ρςG(μ+ν2,y)dy]≤1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)dydx≤14(ν−μ)[∫νμG(x,ς)dx+∫νμG(x,ρ)dx]+14(ρ−ς)[∫ρςG(μ,y)dy+∫ρςG(ν,y)dy]≤G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (4) |
For more details related to inequalities, see [6,7,8,9] and reference therein.
Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore [10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.
Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in [11,12,13,14]. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in [15]. For fuzzy interval-valued function, Khan et al. [16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. [21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] for other related results.
The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.
Let R be the set of real numbers and RI be the space of all closed and bounded intervals of R, such that U∈RI is defined by
U=[U∗,U∗]={y∈R|U∗≤y≤U∗},(U∗,U∗∈R). | (5) |
If U∗=U∗, then U is said to be degenerate. If U∗≥0, then [U∗,U∗] is called positive interval. The set of all positive interval is denoted by R+I and defined as R+I={[U∗,U∗]:[U∗,U∗]∈RIandU∗≥0}.
Let ϱ∈R and ϱU be defined by
ϱ.U={[ϱU∗,ϱU∗]ifϱ>0,{0}ifϱ=0,[ϱU∗,ϱU∗]ifϱ<0. | (6) |
Then, the Minkowski difference D−U, addition U+D and U×D for U,D∈RI are defined by
[D∗,D∗]−[U∗,U∗]=[D∗−U∗,D∗−U∗],[D∗,D∗]+[U∗,U∗]=[D∗+U∗,D∗+U∗], | (7) |
and
[D∗,D∗]×[U∗,U∗]=[min{D∗U∗,D∗U∗,D∗U∗,D∗U∗},max{D∗U∗,D∗U∗,D∗U∗,D∗U∗}]. |
The inclusion "⊇" means that
U⊇D if and only if, [U∗,U∗]⊇[D∗,D∗], and if and only if
U∗≤D∗,D∗≤U∗. | (8) |
Remark 1. [36] (ⅰ) The relation "≤p" is defined on RI by
[D∗,D∗]≤p[U∗,U∗]ifandonlyifD∗≤U∗,D∗≤U∗, | (9) |
for all [D∗,D∗],[U∗,U∗]∈RI, and it is a pseudo order relation. The relation [D∗,D∗]≤p[U∗,U∗] coincident to [D∗,D∗]≤[U∗,U∗] on RI when it is "≤p"
(ⅱ) It can be easily seen that "≤p" looks like "left and right" on the real line R, so we call "≤p" is "left and right" (or "LR" order, in short).
For [D∗,D∗],[U∗,U∗]∈RI, the Hausdorff-Pompeiu distance between intervals [D∗,D∗] and [U∗,U∗] is defined by
d([D∗,D∗],[U∗,U∗])=max{|D∗−U∗|,|D∗−U∗|}. | (10) |
It is familiar fact that (RI,d) is a complete metric space.
Theorem 1. [10] If G:[μ,ν]⊂R→RI is an I-V-F given by (x) [G∗(x),G∗(x)], then G is Riemann integrable over [μ,ν] if and only if, G∗ and G∗ both are Riemann integrable over [μ,ν] such that
(IR)∫νμG(x)dx=[(R)∫νμG∗(x)dx,(R)∫νμG∗(x)dx]. | (11) |
The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by R[μ,ν] and TR[μ,ν], respectively.
Definition 1. [31,33] Let G:[μ,ν]→RI be interval-valued function and G∈TR[μ,ν]. Then interval Riemann-Liouville-type integrals of G are defined as
Iαμ+G(y)=1Γ(α)∫yμ(y−t)α−1G(t)dt(y>μ), | (12) |
Iαν−G(y)=1Γ(α)∫νy(t−y)α−1G(t)dt(y<ν), | (13) |
where α>0 and Γ is the gamma function.
Theorem 2. [20] Let G:[ς,ρ]→RI+ be a LR-convex I-V.F such that G(y)=[G∗(y),G∗(y)] for all y∈[ς,ρ]. If G∈L([ς,ρ],R+I), then
G(ς+ρ2)≤pΓ(α+1)2(ρ−ς)α[Iας+G(ρ)+Iαρ−G(ς)]≤pG(ς)+G(ρ)2. | (14) |
Theorem 3. [20] Let G,S:[ς,ρ]→R+I be two LR-convex I-V.Fs such that G(x)=[G∗(x),G∗(x)] and S(x)=[S∗(x),S∗(x)] for all x∈[ς,ρ]. If G×S∈L([ς,ρ],R+I) is fuzzy Riemann integrable, then
Γ(α+1)2(ρ−ς)α[Iας+G(ρ)×S(ρ)+Iαρ−G(ς)×S(ς)] |
≤p(12−α(α+1)(α+2))M(ς,ρ)+(α(α+1)(α+2))N(ς,ρ), | (15) |
and
G(ς+ρ2)×S(ς+ρ2) |
≤pΓ(α+1)4(ρ−ς)α[Iας+G(ρ)×S(ρ)+Iαρ−G(ς)×S(ς)] |
+12(12−α(α+1)(α+2))M(ς,ρ)+12(α(α+1)(α+2))N(ς,ρ), | (16) |
where M(ς,ρ)=G(ς)×S(ς)+G(ρ)×S(ρ), N(ς,ρ)=G(ς)×S(ρ)+G(ρ)×S(ς),
and M(ς,ρ)=[M∗(ς,ρ),M∗(ς,ρ)] and N(ς,ρ)=[N∗(ς,ρ),N∗(ς,ρ)].
Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted TOΔ, respectively.
Theorem 4. [35] Let Δ=[ς,ρ]×[μ,ν]. If G:Δ→RI is interval-valued doubl integrable (ID-integrable) on Δ. Then, we have
(ID)∫ρς∫νμG(x,y)dydx=(IR)∫ρς(IR)∫νμG(x,y)dydx. |
Definition 2. [36] Let G:Δ→R+I and G∈TOΔ. The interval Riemann-Liouville-type integrals Iα,βμ+,ς+,Iα,βμ+,ρ−, Iα,βν−,ς+,Iα,βν−,ρ− of G order α,β>0 are defined by
Iα,βμ+,ς+G(x,y)=1Γ(α)Γ(β)∫xμ∫yς(x−t)α−1(y−s)β−1G(t,s)dsdt(x>μ,y>ς), | (17) |
Iα,βμ+,ρ−G(x,y)=1Γ(α)Γ(β)∫xμ∫ρy(x−t)α−1(s−y)β−1G(t,s)dsdt(x>μ,y<ρ), | (18) |
Iα,βν−,ς+G(x,y)=1Γ(α)Γ(β)∫νx∫yς(t−x)α−1(y−s)β−1G(t,s)dsdt(x<ν,y>ς), | (19) |
Iα,βν−,ρ−G(x,y)=1Γ(α)Γ(β)∫νx∫ρy(t−x)α−1(s−y)β−1G(t,s)dsdt(x<ν,y<ρ). | (20) |
Definition 3. [38] The I-V.F G:Δ→R+I is said to be coordinated LR-convex I-V.F on Δ if
G(τμ+(1−τ)ν,sς+(1−s)ρ) |
≤pτsG(μ,ς)+τ(1−s)G(μ,ρ)+(1−τ)sG(ν,ς)+(1−τ)(1−s)G(ν,ρ), | (21) |
for all (μ,ν),(ς,ρ)∈Δ, and τ,s∈[0,1]. If inequality (21) is reversed, then G is called coordinate LR-concave I-V.F on Δ.
Lemma 1. [38] Let G:Δ→R+I be an coordinated I-V.F on Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if there exist two coordinated LR-convex I-V.Fs Gx:[ς,ρ]→R+I, Gx(w)=G(x,w) and Gy:[μ,ν]→R+I, Gy(z)=G(z,y).
Theorem 5. [38] Let G:Δ→R+I be a I-V.F on Δ such that
G(x,ϖ)=[G∗(x,ϖ),G∗(x,ϖ)], | (22) |
for all (x,ϖ)∈Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if, G∗(x,ϖ) and G∗(x,ϖ) are coordinated convex functions.
Example 1. We consider the I-V.Fs G:[0,1]×[0,1]→R+I defined by,
G(x)(σ)={σ2(6+ex)(6+eϖ),σ∈[0,2(6+ex)(6+eϖ)]4(6+ex)(6+eϖ)−σ2(6+ex)(6+eϖ),σ∈(2(6+ex)(6+eϖ),4(6+ex)(6+eϖ)]0,otherwise, |
Then, for each θ∈[0,1], we have G(x)=[2θ(6+ex)(6+eϖ),(4+2θ)(6+ex)(6+eϖ)]. Since end point functions G∗((x,ϖ),θ), G∗((x,ϖ),θ) are coordinate concave functions for each θ∈[0,1]. Hence S(x,ϖ) is coordinate LR-concave I-V.F.
From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.
Remark 2. If one takes G∗(x,ϖ)=G∗(x,ϖ), then G is known as coordinated function if G satisfies the coming inequality
G(τμ+(1−τ)ν,sς+(1−s)ρ) |
≤τsG(μ,ς)+τ(1−s)G(μ,ρ)+(1−τ)sG(ν,ς)+(1−τ)(1−s)G(ν,ρ), |
is valid which is defined by Dragomir [5]
Let one takes G∗(x,ϖ)≠G∗(x,ϖ), where G∗(x,ϖ) is affine function and G∗(x,ϖ) is a concave function. If coming inequality,
G(τμ+(1−τ)ν,sς+(1−s)ρ) |
⊇τsG(μ,ς)+τ(1−s)G(μ,ρ)+(1−τ)sG(ν,ς)+(1−τ)(1−s)G(ν,ρ), |
is valid, then G is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]
In this section, we shall continue with the following fractional HH-inequality for coordinated LR-convex I-V.Fs, and we also give fractional HH-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.
Theorem 6. Let G:Δ→R+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G∗(x,y),G∗(x,y)] for all (x,y)∈Δ. If G∈TOΔ, then following inequalities holds:
G(μ+ν2,ς+ρ2)≤pΓ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)] |
≤pΓ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+Iαν−G(μ,ς)+Iαν−G(μ,ρ)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)+Iβρ−G(ν,ς)+Iβς+G(μ,ρ)+Iβρ−G(ν,ς)] |
≤pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (23) |
If G(x) coordinated LR-concave I-V.F, then
G(μ+ν2,ς+ρ2)≥pΓ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)] |
≥pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)] |
≥pΓ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+Iαν−G(μ,ς)+Iαν−G(μ,ρ)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)+Iβρ−G(ν,ς)+Iβς+G(μ,ρ)+Iβρ−G(ν,ς)] |
≥pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (24) |
Proof. Let G:[μ,ν]→R+I be a coordinated LR-convex I-V.F. Then, by hypothesis, we have
4G(μ+ν2,ς+ρ2)≤pG(τμ+(1−τ)ν,τς+(1−τ)ρ)+G((1−τ)μ+τν,(1−τ)ς+τρ). |
By using Theorem 5, we have
4G∗(μ+ν2,ς+ρ2)≤G∗(τμ+(1−τ)ν,τς+(1−τ)ρ)+G∗((1−τ)μ+τν,(1−τ)ς+τρ),4G∗(μ+ν2,ς+ρ2)≤G∗(τμ+(1−τ)ν,τς+(1−τ)ρ)+G∗((1−τ)μ+τν,(1−τ)ς+τρ). |
By using Lemma 1, we have
2G∗(x,ς+ρ2)≤G∗(x,τς+(1−τ)ρ)+G∗(x,(1−τ)ς+τρ),2G∗(x,ς+ρ2)≤G∗(x,τς+(1−τ)ρ)+G∗(x,(1−τ)ς+τρ), | (25) |
and
2G∗(μ+ν2,y)≤G∗(τμ+(1−τ)ν,y)+G∗((1−τ)μ+tν,y),2G∗(μ+ν2,y)≤G∗(τμ+(1−τ)ν,y)+G∗((1−τ)μ+tν,y). | (26) |
From (25) and (26), we have
2[G∗(x,ς+ρ2),G∗(x,ς+ρ2)] |
≤p[G∗(x,τς+(1−τ)ρ),G∗(x,τς+(1−τ)ρ)] |
+[G∗(x,(1−τ)ς+τρ),G∗(x,(1−τ)ς+τρ)], |
and
2[G∗(μ+ν2,y),G∗(μ+ν2,y)] |
≤p[G∗(τμ+(1−τ)ν,y),G∗(τμ+(1−τ)ν,y)] |
+[G∗(τμ+(1−τ)ν,y),G∗(τμ+(1−τ)ν,y)], |
It follows that
G(x,ς+ρ2)≤pG(x,τς+(1−τ)ρ)+G(x,(1−τ)ς+τρ), | (27) |
and
G(μ+ν2,y)≤pG(τμ+(1−τ)ν,y)+G(τμ+(1−τ)ν,y). | (28) |
Since G(x,.) and G(.,y), both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have
Gx(ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+Gx(ρ)+Iβρ−Gx(ς)]≤pGx(ς)+Gx(ρ)2. | (29) |
and
Gy(μ+ν2)≤pΓ(α+1)2(ν−μ)α[Iαμ+Gy(ν)+Iαν−Gy(μ)]≤pGy(μ)+Gy(ν)2 | (30) |
Since Gx(w)=G(x,w), the inequality (29) can be written as
G(x,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iας+G(x,ρ)+Iαρ−G(x,ς)]≤pG(x,ς)+G(x,ρ)2. | (31) |
That is
G(x,ς+ρ2)≤pβ2(ρ−ς)β[∫ρς(ρ−s)β−1G(x,s)ds+∫ρς(s−ς)β−1G(x,s)ds]≤pG(x,ς)+G(x,ρ)2. |
Multiplying double inequality (31) by α(ν−x)α−12(ν−μ)α and integrating with respect to x over [μ,ν], we have
α2(ν−μ)α∫νμG(x,ς+ρ2)(ν−x)α−1dx |
≤p∫νμ∫ρς(ν−x)α−1(ρ−s)β−1G(x,s)dsdx+∫νμ∫ρς(ν−x)α−1(s−ς)β−1G(x,s)dsdx |
≤pα4(ν−μ)α[∫νμ(ν−x)α−1G(x,ς)dx+∫νμ(ν−x)α−1G(x,ρ)dx]. | (32) |
Again multiplying double inequality (31) by α(x−μ)α−12(ν−μ)α and integrating with respect to x over [μ,ν], we have
α2(ν−μ)α∫νμG(x,ς+ρ2)(ν−x)α−1dx |
≤pαβ4(ν−μ)α(ρ−ς)β∫νμ∫ρς(x−μ)α−1(ρ−s)β−1G(x,s)dsdx |
+αβ4(ν−μ)α(ρ−ς)β∫νμ∫ρς(x−μ)α−1(s−ς)β−1G(x,s)dsdx |
≤pα4(ν−μ)α[∫νμ(x−μ)α−1G(x,ς)dx+∫νμ(x−μ)α−1G(x,d)dx]. | (33) |
From (32), we have
Γ(α+1)2(ν−μ)α[Iαμ+G(ν,ς+ρ2)] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν−,ς+G(ν,ς)] |
≤pΓ(α+1)4(ν−μ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)]. | (34) |
From (33), we have
Γ(α+1)2(ν−μ)α[Iαν−G(μ,ς+ρ2)] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)] |
≤pΓ(α+1)4(ν−μ)α[Iαν−G(μ,ς)+Iαν−G(μ,ρ)]. | (35) |
Similarly, since Gy(z)=G(z,y) then, from (34) and (35), (30) we have
Γ(β+1)2(ρ−ς)β[Iβς+G(μ+ν2,ρ)] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν−,ς+G(μ,ρ)] |
≤pΓ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)+Iβς+G(ν,ρ)], | (36) |
and
Γ(β+1)2(ρ−ς)α[Iβρ−G(μ+ν2,ς)] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ρ−G(μ,ς)] |
≤pΓ(β+1)4(ρ−ς)β[Iβρ−G(μ,ς)+Iβρ−G(ν,ς)]. | (37) |
After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).
Now, from left part of inequality (14), we have
G(μ+ν2,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)], | (38) |
and
G(μ+ν2,ς+ρ2)≤pΓ(α+1)2(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)]. | (39) |
Summing the inequalities (38) and (39), we obtain the following inequality:
G(μ+ν2,ς+ρ2) |
≤pΓ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)]+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)], | (40) |
this is the first inequality of (23).
Now, from right part of inequality (14), we have
Γ(β+1)2(ρ−ς)β[Iβς+G(μ,ρ)+Iβρ−G(μ,ς)]≤pG(μ,ς)+G(μ,ρ)2, | (41) |
Γ(β+1)2(ρ−ς)β[Iβς+G(ν,ρ)+Iβρ−G(ν,ς)]≤pG(ν,ς)+G(ν,ρ)2, | (42) |
Γ(α+1)2(ν−μ)α[Iαμ+G(ν,ς)+Iαν−G(μ,ς)]≤pG(μ,ς)+G(ν,ς)2, | (43) |
Γ(α+1)2(ν−μ)α[Iαμ+G(ν,ρ)+Iαν−G(μ,ρ)]≤pG(μ,ρ)+G(ν,ρ)2. | (44) |
Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with 14, we have
Γ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)+Iαν−G(μ,ς)+Iαμ+G(ν,ρ)+Iαν−G(μ,ρ)] |
+Γ(β+1)2(ρ−ς)β[Iβς+G(μ,ρ)+Iβρ−G(μ,ς)+Iβς+G(ν,ρ)+Iβρ−G(ν,ς)] |
≤pG(μ,ς)+G(μ,ρ)+G(ν,ς)+G(ν,ρ)4. | (45) |
This is last inequality of (23) and the result has been proven.
Remark 3. If one to take α=1 and β=1, then from (23), we achieve the coming inequality, see [38]:
G(μ+ν2,ς+ρ2) |
≤p12[1ν−μ∫νμG(x,ς+ρ2)dx+1ρ−ς∫ρςG(μ+ν2,y)dy]≤p1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)dydx≤p14(ν−μ)[∫νμG(x,ς)dx+∫νμG(x,ρ)dx]+14(ρ−ς)[∫ρςG(μ,y)dy+∫ρςG(ν,y)dy] |
≤pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (46) |
Let one takes G∗(x,y) is an affine function and G∗(x,y) is concave function. If G∗(x,y)≠G∗(x,y), then from Remark 2 and (24), we acquire the coming inequality, see [31]:
G(μ+ν2,ς+ρ2)⊇Γ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)]+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)] |
⊇Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)] |
⊇Γ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+Iαν−G(μ,ς)+Iαν−G(μ,ρ)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)˜+Iβρ−G(ν,ς)+Iβς+G(μ,ρ)+Iβρ−G(ν,ς)] |
⊇G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (47) |
Let one takes α=1 and β=1, G∗(x,y) is an affine function and G∗(x,y) is concave function. If G∗(x,y)≠G∗(x,y), then Remark 2 and from (24), we acquire the coming inequality, see [37]:
G(μ+ν2,ς+ρ2) |
⊇12[1ν−μ∫νμG(x,ς+ρ2)dx+1ρ−ς∫ρςG(μ+ν2,y)dy]⊇1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)dydx |
⊇14(ν−μ)[∫νμG(x,ς)dx+∫νμG(x,ρ)dx]+14(ρ−ς)[∫ρςG(μ,y)dy+∫ρςG(ν,y)dy] |
⊇G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. | (48) |
Example 2. We consider the I-V-Fs G:[0,1]×[0,1]→R+I defined by,
G(x)=[2,6](6+ex)(6+ey). |
Since end point functions G∗(x,y), G∗(x,y) are convex functions on coordinate, then G(x,y) is convex I-V-F on coordinate. Then for α=1 and β=1, we have
G(μ+ν2,ς+ρ2)=[2(5+e12)2,6(6+e12)2], |
Γ(α+1)4(ν−μ)α[Iαμ+G(ν,ς+ρ2)+Iαν−G(μ,ς+ρ2)]+Γ(β+1)4(ρ−ς)β[Iβς+G(μ+ν2,ρ)+Iβρ−G(μ+ν2,ς)] |
=[4(6+e12)(5+e),12(6+e12)(5+e)], |
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)+Iα,βν−,ς+G(μ,ρ)+Iα,βν−,ρ−G(μ,ς)] |
=[2(5+e)2,6(5+e)2], |
Γ(α+1)8(ν−μ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+Iαν−G(μ,ς)+Iαν−G(μ,ρ)] |
+Γ(β+1)4(ρ−ς)β[Iβς+G(μ,ρ)˜+Iβρ−G(ν,ς)+Iβς+G(μ,ρ)+Iβρ−G(ν,ς)] |
=[(5+e)(13+e),3(5+e)(13+e)] |
G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4=[(6+e)(20+e)+492,6((6+e)(20+e)+49)2]. |
That is
[2(5+e12)2,6(6+e12)2]≤p[4(6+e12)(5+e),12(6+e12)(5+e)] |
≤p[2(5+e)2,6(5+e)2] |
≤p[(5+e)(13+e),3(5+e)(13+e)] |
≤p[(6+e)(20+e)+492,3((6+e)(20+e)+49)]. |
Hence, Theorem 3.1 has been verified
Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs
Theorem 7. Let G,S:Δ→R+I be a coordinate LR-convex I-V.Fs on Δ such that G(x,y)=[G∗(x,y),G∗(x,y)] and S(x,y)=[S∗(x,y),S∗(x,y)] for all (x,y)∈Δ. If G×S∈TOΔ, then following inequalities holds:
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≤p(12−α(α+1)(α+2))(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12−β(β+1)(β+2))L(μ,ν,ς,ρ) |
+(12−α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). | (49) |
If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≥p(12−α(α+1)(α+2))(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12−β(β+1)(β+2))L(μ,ν,ς,ρ) |
+(12−α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). | (50) |
Where
K(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)+G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ), |
L(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ς)˜+G(ν,ρ)×S(μ,ρ)+G(ν,ς)×S(μ,ς)+G(μ,ρ)×S(ν,ρ), |
M(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)+G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς), |
N(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)+G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς). |
and K(μ,ν,ς,ρ), ˜L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are defined as follows:
K(μ,ν,ς,ρ)=[K∗(μ,ν,ς,ρ),K∗(μ,ν,ς,ρ)], |
L(μ,ν,ς,ρ)=[L∗(μ,ν,ς,ρ),L∗(μ,ν,ς,ρ)], |
M(μ,ν,ς,ρ)=[M∗(μ,ν,ς,ρ),M∗(μ,ν,ς,ρ)], |
N(μ,ν,ς,ρ)=[N∗(μ,ν,ς,ρ),N∗(μ,ν,ς,ρ)]. |
Proof. Let G and S both are coordinated LR-convex I-V.Fs on [μ,ν]×[ς,ρ]. Then
G(τμ+(1−τ)ν,sς+(1−s)ρ) |
≤pτsG(μ,ς)+τ(1−s)G(μ,ρ)+(1−τ)sG(ν,ς)+(1−τ)(1−s)G(ν,ρ), |
and
S(τμ+(1−τ)ν,sς+(1−s)ρ) |
≤pτsS(μ,ς)+τ(1−s)S(μ,ρ)+(1−τ)sS(ν,ς)+(1−τ)(1−s)S(ν,ρ). |
Since G and S both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist
Gx:[ς,ρ]→R+I,Gx(y)=G(x,y),Sx:[ς,ρ]→R+I,Sx(y)=S(x,y), |
Since Gx, and Sx are I-V.Fs, then by inequality (15), we have
Γ(β+1)2(ρ−ς)β[Iβς+Gx(ρ)×Sx(ρ)+Iβρ−Gx(ς)×Sx(ς)] |
≤p(12−β(β+1)(β+2))(Gx(ς)×Sx(ς)+Gx(ρ)×Sx(ρ)) |
+(β(β+1)(β+2))(Gx(ς)×Sx(ρ)+Gx(ρ)×Sx(ς)). |
That is
β2(ρ−ς)β[∫ρς(ρ−y)β−1G(x,y)×S(x,y)ρy+∫ρς(y−ς)β−1G(x,y)×S(x,y)ρy] |
≤p(12−β(β+1)(β+2))(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ)) |
+(β(β+1)(β+2))(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς)). | (51) |
Multiplying double inequality (51) by α(ν−x)α−12(ν−μ)α and integrating with respect to x over [μ,ν], we get
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)] |
≤pΓ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ)) |
+Γ(α+1)2(ν−μ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). | (52) |
Again, multiplying double inequality (51) by α(x−μ)α−12(ν−μ)α and integrating with respect to x over [μ,ν], we gain
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≤pΓ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαν−G(μ,ς)×S(μ,ς)+Iαν−G(μ,ρ)×S(μ,ρ)) |
+Γ(α+1)2(ν−μ)αβ(β+1)(β+2)(Iαν−G(μ,ς)×S(μ,ρ)+Iαν−G(μ,ρ)×S(μ,ς)). | (53) |
Summing (52) and (53), we have
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≤pΓ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαν−G(μ,ς)×S(μ,ς)) |
+Γ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαμ+G(ν,ρ)×S(ν,ρ)+Iαν−G(μ,ρ)×S(μ,ρ)) |
+Γ(α+1)2(ν−μ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαν−G(μ,ς)×S(μ,ρ)) |
+Γ(α+1)2(ν−μ)αβ(β+1)(β+2)(Iαμ+G(ν,ρ)×S(ν,ς)+Iαν−G(μ,ρ)×S(μ,ς)). | (54) |
Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation
Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαν−G(μ,ς)×S(μ,ς)) |
≤p(12−α(α+1)(α+2))(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)) |
+(α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)). | (55) |
Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ρ)×S(ν,ρ)+Iαν−G(μ,ρ)×S(μ,ρ)) |
≤p(12−α(α+1)(α+2))(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ)) |
+(α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)). | (56) |
Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαν−G(μ,ς)×S(μ,ρ)) |
≤p(12−α(α+1)(α+2))(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)) |
+(α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)). | (57) |
And
Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ρ)×S(ν,ς)+Iαν−G(μ,ρ)×S(μ,ς)) |
≤p(12−α(α+1)(α+2))(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς)) |
+(α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)). | (58) |
From (55)–(58), inequality (54) we have
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≤p(12−α(α+1)(α+2))(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12−β(β+1)(β+2))L(μ,ν,ς,ρ) |
+(12−α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). |
Hence, the result has been proven.
Remark 4. If one to take α=1 and β=1, then from (49), we achieve the coming inequality, see [38]:
1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)×S(x,y)dydx |
≤p19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). | (59) |
Let one takes G∗(x,y) is an affine function and G∗(x,y) is concave function. If G∗(x,y)≠G∗(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [36]:
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
⊇(12−α(α+1)(α+2))(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12−β(β+1)(β+2))L(μ,ν,ς,ρ) |
+(12−α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). | (60) |
Let one takes G∗(x,y) is an affine function and G∗(x,y) is concave function. If G∗(x,y)≠G∗(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [37]:
1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)×S(x,y)dydx |
⊇19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). | (61) |
If G∗(x,y)=G∗(x,y) and S∗(x,y)=S∗(x,y), then from (49), we acquire the coming inequality, see [39]:
Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)] |
≤(12−α(α+1)(α+2))(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12−β(β+1)(β+2))L(μ,ν,ς,ρ) |
+(12−α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). | (62) |
Theorem 8. Let G,S:Δ→R+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G∗(x,y),G∗(x,y)] and S(x,y)=[S∗(x,y),S∗(x,y)] for all (x,y)∈Δ. If G×S∈TOΔ, then following inequalities holds:
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2) |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12−α(α+1)(α+2))]K(μ,ν,ς,ρ) |
+[12(12−α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ) |
+[12(12−β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ) |
+[14−α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). | (63) |
If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≥pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12−α(α+1)(α+2))]K(μ,ν,ς,ρ) |
+[12(12−α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12−β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14−α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). | (64) |
Where K(μ,ν,ς,ρ), L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are given in Theorem 7.
Proof. Since G,S:Δ→R+I be two LR-convex I-V.Fs, then from inequality (16), we have
2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤pα2(ν−μ)α[∫νμ(ν−x)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx+∫νμ(x−μ)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+(α(α+1)(α+2))(G(μ,ς+ρ2)×S(μ,ς+ρ2)+G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12−α(α+1)(α+2))(G(μ,ς+ρ2)×S(ν,ς+ρ2)+G(ν,ς+ρ2)×S(μ,ς+ρ2)), | (65) |
and
2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤pβ2(ρ−ς)β[∫ρς(ρ−y)β−1G(μ+ν2,y)×S(μ+ν2,y)dy+∫ρς(y−ς)β−1G(μ+ν2,y)×S(μ+ν2,y)dy]+(β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ς)+G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12−β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ρ)+G(μ+ν2,ρ)×S(μ+ν2,ς)), | (66) |
Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain
8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤pα2(ν−μ)α[∫νμ2(ν−x)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx+∫νμ2(x−μ)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+β2(ρ−ς)β[∫ρς2(ρ−y)β−1G(μ+ν2,y)×S(μ+ν2,y)dy+∫ρς2(y−ς)β−1G(μ+ν2,y)×S(μ+ν2,y)dy]+(α(α+1)(α+2))(2G(μ,ς+ρ2)×S(μ,ς+ρ2)+2G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12−α(α+1)(α+2))(2G(μ,ς+ρ2)×S(ν,ς+ρ2)+2G(ν,ς+ρ2)×S(μ,ς+ρ2))+(β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ς)+2G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12−β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ρ)+2G(μ+ν2,ρ)×S(μ+ν2,ς)). | (67) |
Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have
α2(ν−μ)α∫νμ2(ν−x)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx≤pαβ4(ν−μ)α(ρ−ς)β[∫νμ∫ρς(ν−x)α−1(ρ−y)β−1G(x,y)dydx+∫νμ∫ρς(ν−x)α−1(y−ς)β−1G(x,y)dydx]+β(β+1)(β+2)α2(ν−μ)α∫νμ(ν−x)α−1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12−β(β+1)(β+2))α2(ν−μ)α∫νμ(ν−x)α−1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)]+Γ(α+1)2(ν−μ)α(β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). | (68) |
α2(ν−μ)α∫νμ2(x−μ)α−1G(x,ς+ρ2)×S(x,ς+ρ2)dx≤pαβ4(ν−μ)α(ρ−ς)β[∫νμ∫ρς(x−μ)α−1(ρ−y)β−1G(x,y)dydx+∫νμ∫ρς(x−μ)α−1(y−ς)β−1G(x,y)dydx]+β(β+1)(β+2)α2(ν−μ)α∫νμ(x−μ)α−1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12−β(β+1)(β+2))α2(ν−μ)α∫νμ(x−μ)α−1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+Γ(α+1)2(ν−μ)α(β(β+1)(β+2))(Iαν−G(μ,ς)×S(μ,ς)+Iαν−G(μ,ρ)×S(μ,ρ))+Γ(α+1)2(ν−μ)α(12−β(β+1)(β+2))(Iαν−G(μ,ς)×S(μ,ρ)+Iαν−G(μ,ρ)×S(μ,ς)). | (69) |
β2(ρ−ς)β[∫ρς2(ρ−y)β−1G(μ+ν2,y)×S(μ+ν2,y)dy] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)]+Γ(β+1)2(ρ−ς)β(α(α+1)(α+2))(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρ−ς)β(12−α(α+1)(α+2))(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(ν,ρ)). | (70) |
β2(ρ−ς)β[∫ρς2(y−ς)β−1G(μ+ν2,y)×S(μ+ν2,y)dy] |
≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ρ−G(ν,ς)×S(ν,ς)]+Γ(β+1)2(ρ−ς)β(α(α+1)(α+2))(Iβρ−G(μ,ς)×S(μ,ς)+Iβρ−G(ν,ς)×S(ν,ς))+Γ(β+1)2(ρ−ς)β(12−α(α+1)(α+2))(Iβρ−G(μ,ς)×S(ν,ς)+Iβρ−G(ν,ς)×S(ν,ς)). | (71) |
And
2G(μ+ν2,ς)×S(μ+ν2,ς)≤pΓ(α+1)2(ν−μ)α[Iαμ+G(ν,ς)×S(ν,ς)+Iαν−G(μ,ς)×S(μ,ς)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))+(12−α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)), | (72) |
2G(μ+ν2,ρ)×S(μ+ν2,ρ)≤pΓ(α+1)2(ν−μ)α[Iαμ+G(ν,ρ)×S(ν,ρ)+Iαν−G(μ,ρ)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))+(12−α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)), | (73) |
2G(μ+ν2,ς)×S(μ+ν2,ρ)≤pΓ(α+1)2(ν−μ)α[Iαμ+G(ν,ς)×S(ν,ρ)+Iαν−G(μ,ς)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))+(12−α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)), | (74) |
2G(μ+ν2,ρ)×S(μ+ν2,ς)≤pΓ(α+1)2(ν−μ)α[Iαμ+G(ν,ρ)×S(ν,ς)+Iαν−G(μ,ρ)×S(μ,ς)] |
+α(α+1)(α+2)(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))+(12−α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)), | (75) |
2G(μ,ς+ρ2)×S(μ,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+G(μ,ρ)×S(μ,ρ)+Iβρ−G(μ,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(μ,ς)×S(μ,ς)+G(μ,ρ)×S(μ,ρ))+(12−β(β+1)(β+2))(G(μ,ς)×S(μ,ρ)+G(μ,ρ)×S(μ,ς)), | (76) |
2G(ν,ς+ρ2)×Sϕ(ν,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+G(ν,ρ)×S(ν,ρ)+Iβρ−G(ν,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(ν,ς)×S(ν,ς)+G(ν,ρ)×S(ν,ρ))+(12−β(β+1)(β+2))(G(ν,ς)×S(ν,ρ)+G(ν,ρ)×S(ν,ς)), | (77) |
2G(μ,ς+ρ2)×S(ν,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+G(μ,ρ)×S(ν,ρ)+Iβρ−G(μ,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(μ,ς)×S(ν,ς)+G(μ,ρ)×S(ν,ρ))+(12−β(β+1)(β+2))(G(μ,ς)×S(ν,ρ)+G(μ,ρ)×S(ν,ς)), | (78) |
and
2G(ν,ς+ρ2)×S(μ,ς+ρ2)≤pΓ(β+1)2(ρ−ς)β[Iβς+G(ν,ρ)×S(μ,ρ)+Iβρ−G(ν,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(ν,ς)×S(μ,ς)+G(ν,ρ)×S(μ,ρ))+(12−β(β+1)(β+2))(G(ν,ς)×S(μ,ρ)+G(ν,ρ)×S(μ,ς)), | (79) |
From inequalities (68) to (79), inequality (67) we have
8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤pΓ(α+1)Γ(β+1)2(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+(2α(α+1)(α+2))[Γ(β+1)2(ρ−ς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρ−ς)β(Iβρ−G(μ,ς)×S(μ,ς)+Iβρ−G(ν,ς)×S(ν,ς))]+2(12−α(α+1)(α+2))[Γ(β+1)2(ρ−ς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρ−ς)β(Iβρ−G(μ,ς)×S(ν,ς)+Iβρ−G(ν,ς)×S(μ,ς))]+2(β(β+1)(β+2))[Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(ν−μ)α(Iαν−G(μ,ς)×S(μ,ς)+Iαν−G(μ,ρ)×S(μ,ρ))]+2(12−β(β+1)(β+2))[Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς))+Γ(α+1)2(ν−μ)α(Iαν−G(μ,ς)×S(μ,ρ)+Iαν−G(μ,ρ)×S(μ,ς))] |
+2α(α+1)(α+2)β(β+1)(β+2)K(μ,ν,ς,ρ)++(12−α(α+1)(α+2))2β(β+1)(β+2)L(μ,ν,ς,ρ) |
+2α(α+1)(α+2)(12−β(β+1)(β+2))M(μ,ν,ς,ρ)+2(12−α(α+1)(α+2))(12−β(β+1)(β+2))N(μ,ν,ς,ρ). | (80) |
Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have
Γ(β+1)2(ρ−ς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρ−ς)β(Iβρ−G(μ,ς)×S(μ,ς)+Iβρ−G(ν,ς)×S(ν,ς))≤p(12−β(β+1)(β+2))K(μ,ν,ς,ρ)+β(β+1)(β+2)M(μ,ν,ς,ρ). | (81) |
Γ(β+1)2(ρ−ς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρ−ς)β(Iβρ−G(μ,ς)×S(ν,ς)+Iβρ−G(ν,ς)×S(μ,ς))≤p(12−β(β+1)(β+2))L(μ,ν,ς,ρ)+β(β+1)(β+2)N(μ,ν,ς,ρ). | (82) |
Γ(α+1)2(ν−μ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(ν−μ)α(Iαν−G(μ,ς)×S(μ,ς)+Iαν−G(μ,ρ)×S(μ,ρ))≤p(12−α(α+1)(α+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)L(μ,ν,ς,ρ). | (83) |
Γ(α+1)2(ν−μ)α(Iαν−G(μ,ς)×S(μ,ρ)+Iαν−G(μ,ρ)×S(μ,ς))+Γ(α+1)2(ν−μ)α(Iαν−G(μ,ς)×S(μ,ρ)+Iαν−G(μ,ρ)×S(μ,ς))≤p(12−α(α+1)(α+2))M(μ,ν,ς,ρ)+α(α+1)(α+2)N(μ,ν,ς,ρ). | (84) |
From (77) to (84), (80) we have
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤pΓ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12−α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12−α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12−β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14−α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). | (85) |
This concludes the proof of Theorem 8 result has been proven.
Remark 5. If we take α=1 and β=1, then from (63), we achieve the coming inequality, see [38]:
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤p1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). | (86) |
Let one takes G∗(x,y) is an affine function and G∗(x,y) is convex function. If G∗(x,y)≠G∗(x,y), then from Remark 2 and (64), we acquire the coming inequality, see [37]:
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)⊇1(ν−μ)(ρ−ς)∫νμ∫ρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). | (87) |
Let one takes G∗(x,y) is an affine function and G∗(x,y) is convex function. If G∗(x,y)≠G∗(x,y), then from Remark 2 and (64) we acquire the coming inequality, see [36]:
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)⊇Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12−α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12−α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12−β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14−α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). | (88) |
If we take G∗(x,y)=G∗(x,y) and S∗(x,y)=S∗(x,y), then from (63), we acquire the coming inequality, see [39]:
4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)≤Γ(α+1)Γ(β+1)4(ν−μ)α(ρ−ς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρ−G(ν,ς)×S(ν,ς)+Iα,βν−,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν−,ρ−G(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12−α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12−α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12−β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14−α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). | (89) |
In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.
The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.
The authors declare that they have no competing interests.
[1] | Armstrong N (2007) Paediatric exercise physiology: advances in sport and exercise science series. Churchill Livingstone. |
[2] |
Lee IM, Shiroma EJ, Lobelo F, et al. (2012) Effect of physical inactivity on major non-communicable diseases worldwide: An analysis of burden of disease and life expectancy. Lancet 380: 219-229. https://doi.org/10.1016/S0140-6736(12)61031-9 ![]() |
[3] |
Ahlskog JE (2018) Aerobic exercise: evidence for a direct brain effect to slow parkinson disease progression. Mayo Clin Proc 93: 360-372. https://doi.org/10.1016/j.mayocp.2017.12.015 ![]() |
[4] |
Jakowec MW, Wang Z, Holschneider D, et al. (2016) Engaging cognitive circuits to promote motor recovery in degenerative disorders. exercise as a learning modality. J Hum Kinet 52: 35-51. https://doi.org/10.1515/hukin-2015-0192 ![]() |
[5] |
Preston N, Magallón S, Hill LJB, et al. (2017) A systematic review of high quality randomized controlled trials investigating motor skill programmes for children with developmental coordination disorder. Clin Rehabil 31: 857-870. https://doi.org/10.1177/0269215516661014 ![]() |
[6] | Smits-Engelsman BCM, Jelsma LD, Ferguson GD, et al. (2015) Motor learning: An analysis of 100 trials of a ski slalom game in children with and without developmental coordination disorder. PLoS One 10: 1-19. https://doi.org/10.1371/journal.pone.0140470 |
[7] |
Piepmeier AT, Shih CH, Whedon M, et al. (2015) The effect of acute exercise on cognitive performance in children with and without ADHD. J Sport Heal Sci 4: 97-104. https://doi.org/10.1016/j.jshs.2014.11.004 ![]() |
[8] |
Medina JA, Netto TLB, Muszkat M, et al. (2010) Exercise impact on sustained attention of ADHD children, methylphenidate effects. ADHD Atten Def Hyp Disord 2: 49-58. https://doi.org/10.1007/s12402-009-0018-y ![]() |
[9] |
Saadati H, Esmaeili-Mahani S, Esmaeilpour K, et al. (2015) Exercise improves learning and memory impairments in sleep deprived female rats. Physiol Behav 138: 285-291. https://doi.org/10.1016/j.physbeh.2014.10.006 ![]() |
[10] |
Stathopoulou G, Powers MB, Berry A, et al. (2006) Exercise interventions for mental health: A quantitative and qualitative review. Clin Psychol Sci Pract 13: 179-193. https://doi.org/10.1111/j.1468-2850.2006.00021.x ![]() |
[11] |
Erickson KI, Hillman C, Stillman CM, et al. (2019) Physical activity, cognition, and brain outcomes: A review of the 2018 physical activity guidelines. Med Sci Sport Exerc 51: 1242-1251. https://doi.org/10.1249/MSS.0000000000001936 ![]() |
[12] |
Haverkamp BF, Wiersma R, Vertessen K, et al. (2020) Effects of physical activity interventions on cognitive outcomes and academic performance in adolescents and young adults: A meta-analysis. J Sports Sci 38: 2637-2660. https://doi.org/10.1080/02640414.2020.1794763 ![]() |
[13] |
Stillman CM, Esteban-Cornejo I, Brown B, et al. (2020) Effects of exercise on brain and cognition across age groups and health states. Trends Neurosci 43: 533-543. https://doi.org/10.1016/j.tins.2020.04.010 ![]() |
[14] |
Voss MW, Nagamatsu LS, Liu-Ambrose T, et al. (2011) Exercise, brain, and cognition across the life span. J Appl Physiol 111: 1505-1513. https://doi.org/10.1152/japplphysiol.00210.2011 ![]() |
[15] |
Best JR (2010) Effects of physical activity on children's executive function: contributions of experimental research on aerobic exercise. Dev Rev 30: 331-351. https://doi.org/10.1016/j.dr.2010.08.001 ![]() |
[16] |
de Greeff JW, Bosker RJ, Oosterlaan J, et al. (2018) Effects of physical activity on executive functions, attention and academic performance in preadolescent children: a meta-analysis. J Sci Med Sport 21: 501-507. https://doi.org/10.1016/j.jsams.2017.09.595 ![]() |
[17] |
Smith PJ, Blumenthal JA, Hoffman BM, et al. (2010) Aerobic exercise and neurocognitive performance: A meta-analytic review of randomized controlled trials. Psychosom Med 72: 239-252. https://doi.org/10.1097/PSY.0b013e3181d14633 ![]() |
[18] |
Tsukamoto H, Suga T, Takenaka S, et al. (2016) Greater impact of acute high-intensity interval exercise on post-exercise executive function compared to moderate-intensity continuous exercise. Physiol Behav 155: 224-230. https://doi.org/10.1016/j.physbeh.2015.12.021 ![]() |
[19] |
Griffin ÉW, Mullally S, Foley C, et al. (2011) Aerobic exercise improves hippocampal function and increases BDNF in the serum of young adult males. Physiol Behav 104: 934-941. https://doi.org/10.1016/j.physbeh.2011.06.005 ![]() |
[20] |
Tomporowski PD, Lambourne K, Okumura MS (2011) Physical activity interventions and children's mental function: an introduction and overview. Prev Med 52: S3-S9. https://doi.org/10.1016/j.ypmed.2011.01.028 ![]() |
[21] |
Pereira AC, Huddleston DE, Brickman AM, et al. (2007) An in vivo correlate of exercise-induced neurogenesis in the adult dentate gyrus. Proc Natl Acad Sci USA 104: 5638-5643. https://doi.org/10.1073/pnas.0611721104 ![]() |
[22] |
Yanagisawa H, Dan I, Tsuzuki D, et al. (2010) Acute moderate exercise elicits increased dorsolateral prefrontal activation and improves cognitive performance with Stroop test. Neuroimage 50: 1702-1710. https://doi.org/10.1016/j.neuroimage.2009.12.023 ![]() |
[23] |
Martínez-Drudis L, Amorós-Aguilar L, Torras-Garcia M, et al. (2021) Delayed voluntary physical exercise restores “when” and “where” object recognition memory after traumatic brain injury. Behav Brain Res 400: 113048. https://doi.org/10.1016/j.bbr.2020.113048 ![]() |
[24] |
Skriver K, Roig M, Lundbye-Jensen J, et al. (2014) Acute exercise improves motor memory: Exploring potential biomarkers. Neurobiol Learn Mem 116: 46-58. https://doi.org/10.1016/j.nlm.2014.08.004 ![]() |
[25] |
Vaynman S, Ying Z, Gomez-Pinilla F (2004) Hippocampal BDNF mediates the efficacy of exercise on synaptic plasticity and cognition. Eur J Neurosci 20: 2580-2590. https://doi.org/10.1111/j.1460-9568.2004.03720.x ![]() |
[26] | Chieffi S, Messina G, Villano I, et al. (2017) Neuroprotective effects of physical activity: Evidence from human and animal studies. Front Neurol 8: 1-7. https://doi.org/10.3389/fneur.2017.00188 |
[27] |
Klein C, Rasińska J, Empl L, et al. (2016) Physical exercise counteracts MPTP-induced changes in neural precursor cell proliferation in the hippocampus and restores spatial learning but not memory performance in the water maze. Behav Brain Res 307: 227-238. https://doi.org/10.1016/j.bbr.2016.02.040 ![]() |
[28] |
Neeper SA, Gómez-Pinilla F, Choi J, et al. (1996) Physical activity increases mRNA for brain-derived neurotrophic factor and nerve growth factor in rat brain. Brain Res 726: 49-56. https://doi.org/10.1016/0006-8993(96)00273-9 ![]() |
[29] |
Nokia MS, Lensu S, Ahtiainen JP, et al. (2016) Physical exercise increases adult hippocampal neurogenesis in male rats provided it is aerobic and sustained. J Physiol 594: 1855-1873. https://doi.org/10.1113/JP271552 ![]() |
[30] |
Shors TJ (2009) Saving new brain cells. Sci Am 300: 46-52. ![]() |
[31] |
van Praag H, Shubert T, Zhao C, et al. (2005) Exercise enhances learning and hippocampal neurogenesis in aged mice. J Neurosci 25: 8680-8685. https://doi.org/10.1523/JNEUROSCI.1731-05.2005 ![]() |
[32] |
van Praag H, Christie BR, Sejnowski TJ, et al. (1999) Running enhances neurogenesis, learning, and long-term potentiation in mice. Proc Natl Acad Sci USA 96: 13427-13431. https://doi.org/10.1073/pnas.96.23.13427 ![]() |
[33] |
Ding Q, Vaynman S, Akhavan M, et al. (2006) Insulin-like growth factor I interfaces with brain-derived neurotrophic factor-mediated synaptic plasticity to modulate aspects of exercise-induced cognitive function. Neuroscience 140: 823-833. https://doi.org/10.1016/j.neuroscience.2006.02.084 ![]() |
[34] |
da Costa Daniele TM, de Bruin PFC, de Matos RS, et al. (2020) Exercise effects on brain and behavior in healthy mice, Alzheimer's disease and Parkinson's disease model—A systematic review and meta-analysis. Behav Brain Res 383: 112488. https://doi.org/10.1016/j.bbr.2020.112488 ![]() |
[35] |
Diederich K, Bastl A, Wersching H, et al. (2017) Effects of different exercise strategies and intensities on memory performance and neurogenesis. Front Behav Neurosci 11: 1-9. https://doi.org/10.3389/fnbeh.2017.00047 ![]() |
[36] |
Gutierrez RMS, Ricci NA, Gomes QRS, et al. (2018) The effects of acrobatic exercise on brain plasticity: a systematic review of animal studies. Brain Struct Funct 223: 2055-2071. https://doi.org/10.1007/s00429-018-1631-3 ![]() |
[37] |
Kim TW, Park HS (2018) Physical exercise improves cognitive function by enhancing hippocampal neurogenesis and inhibiting apoptosis in male offspring born to obese mother. Behav Brain Res 347: 360-367. https://doi.org/10.1016/j.bbr.2018.03.018 ![]() |
[38] |
Snigdha S, de Rivera C, Milgram NW, et al. (2014) Exercise enhances memory consolidation in the aging brain. Front Aging Neurosci 6: 1-14. https://doi.org/10.3389/fnagi.2014.00003 ![]() |
[39] |
Loprinzi PD, Frith E (2019) Protective and therapeutic effects of exercise on stress-induced memory impairment. J Physiol Sci 69: 1-12. https://doi.org/10.1007/s12576-018-0638-0 ![]() |
[40] |
Fuss J, Biedermann SV, Falfán-Melgoza C, et al. (2014) Exercise boosts hippocampal volume by preventing early age-related gray matter loss. Hippocampus 24: 131-134. https://doi.org/10.1002/hipo.22227 ![]() |
[41] |
Real CC, Garcia PC, Britto LRG, et al. (2015) Different protocols of treadmill exercise induce distinct neuroplastic effects in rat brain motor areas. Brain Res 1624: 188-198. https://doi.org/10.1016/j.brainres.2015.06.052 ![]() |
[42] |
Salame S, Garcia PC, Real CC, et al. (2016) Distinct neuroplasticity processes are induced by different periods of acrobatic exercise training. Behav Brain Res 308: 64-74. https://doi.org/10.1016/j.bbr.2016.04.029 ![]() |
[43] |
Modaberi S, Shahbazi M, Dehghan M, et al. (2018) The role of mild treadmill exercise on spatial learning and memory and motor activity in animal models of ibotenic acid-induced striatum lesion. Sport Sci Health 14: 587-596. https://doi.org/10.1007/s11332-018-0467-9 ![]() |
[44] |
Vaynman S, Gomez-pinilla F (2006) Revenge of the “sit”: How lifestyle impacts neuronal and cognitive health through molecular systems that interface energy metabolism with neuronal plasticity. J Neurosci Res 84: 699-715. https://doi.org/10.1002/jnr.20979 ![]() |
[45] |
Swain RA, Berggren KL, Kerr AL, et al. (2012) On aerobic exercise and behavioral and neural plasticity. Brain Sci 2: 709-744. https://doi.org/10.3390/brainsci2040709 ![]() |
[46] |
El-Sayes J, Harasym D, Turco CV, et al. (2019) Exercise-induced neuroplasticity: a mechanistic model and prospects for promoting plasticity. Neuroscientist 25: 65-85. https://doi.org/10.1177/1073858418771538 ![]() |
[47] |
Feter N, Alt R, Dias MG, et al. (2019) How do different physical exercise parameters modulate brain-derived neurotrophic factor in healthy and non-healthy adults? A systematic review, meta-analysis and meta-regression. Sci Sport 34: 293-304. https://doi.org/10.1016/j.scispo.2019.02.001 ![]() |
[48] |
Cotman CW, Berchtold NC, Christie LA (2007) Exercise builds brain health: key roles of growth factor cascades and inflammation. Trends Neurosci 30: 464-472. https://doi.org/10.1016/j.tins.2007.06.011 ![]() |
[49] |
Voss MW, Erickson KI, Prakash RS, et al. (2013) Neurobiological markers of exercise-related brain plasticity in older adults. Brain Behav Immun 28: 90-99. https://doi.org/10.1016/j.bbi.2012.10.021 ![]() |
[50] |
Erickson KI, Prakash RS, Voss MW, et al. (2009) Aerobic fitness is associated with hippocampal volume in elderly humans. Hippocampus 19: 1030-1039. https://doi.org/10.1002/hipo.20547 ![]() |
[51] |
Erickson KI, Voss MW, Prakash RS, et al. (2011) Exercise training increases size of hippocampus and improves memory. Proc Natl Acad Sci USA 108: 3017-3022. https://doi.org/10.1073/pnas.1015950108 ![]() |
[52] |
Feter N, Penny JC, Freitas MP, et al. (2018) Effect of physical exercise on hippocampal volume in adults: Systematic review and meta-analysis. Sci Sport 33: 327-338. https://doi.org/10.1016/j.scispo.2018.02.011 ![]() |
[53] | Becker L, Kutz D, Voelcker-Rehage C (2016) Exercise-induced changes in basal ganglia volume and their relation to cognitive performance. J Neurol Neuromedicine 1: 19-24. https://doi.org/10.29245/2572.942x/2016/5.1044 |
[54] |
McMorris T, Hale BJ (2012) Differential effects of differing intensities of acute exercise on speed and accuracy of cognition: A meta-analytical investigation. Brain Cogn 80: 338-351. https://doi.org/10.1016/j.bandc.2012.09.001 ![]() |
[55] |
Roig M, Nordbrandt S, Geertsen SS, et al. (2013) The effects of cardiovascular exercise on human memory: A review with meta-analysis. Neurosci Biobehav Rev 37: 1645-1666. https://doi.org/10.1016/j.neubiorev.2013.06.012 ![]() |
[56] |
de Sousa AFM, Medeiros AR, Del Rosso S, et al. (2019) The influence of exercise and physical fitness status on attention: a systematic review. Int Rev Sport Exercise Psychol 12: 202-234. https://doi.org/10.1080/1750984X.2018.1455889 ![]() |
[57] |
Cotman CW, Berchtold NC (2002) Exercise: A behavioral intervention to enhance brain health and plasticity. Trends Neurosci 25: 295-301. https://doi.org/10.1016/S0166-2236(02)02143-4 ![]() |
[58] | Hasan SMM, Rancourt SN, Austin MW, et al. (2016) Defining optimal aerobic exercise parameters to affect complex motor and cognitive outcomes after stroke: a systematic review and synthesis. Neural Plast . https://doi.org/10.1155/2016/2961573 |
[59] |
Singh AM, Neva JL, Staines WR (2016) Aerobic exercise enhances neural correlates of motor skill learning. Behav Brain Res 301: 19-26. https://doi.org/10.1016/j.bbr.2015.12.020 ![]() |
[60] |
Salthouse TA, Davis HP (2006) Organization of cognitive abilities and neuropsychological variables across the lifespan. Dev Rev 26: 31-54. https://doi.org/10.1016/j.dr.2005.09.001 ![]() |
[61] |
Hillman CH, Erickson KI, Kramer AF (2008) Be smart, exercise your heart: exercise effects on brain and cognition. Nat Rev Neurosci 9: 58-65. https://doi.org/10.1038/nrn2298 ![]() |
[62] |
Castelli DM, Hillman CH, Buck SM, et al. (2007) Physical fitness and academic achievement in third- and fifth-grade students. J Sport Exerc Psychol 29: 239-252. https://doi.org/10.1123/jsep.29.2.239 ![]() |
[63] |
Fedewa AL, Ahn S (2011) The effects of physical activity and physical fitness on children's achievement and cognitive outcomes:a meta-analysis. Res Q Exerc Sport 82: 521-535. https://doi.org/10.1080/02701367.2011.10599785 ![]() |
[64] |
Kantomaa MT, Stamatakis E, Kankaanpää A, et al. (2013) Physical activity and obesity mediate the association between childhood motor function and adolescents' academic achievement. Proc Natl Acad Sci USA 110: 1917-1922. https://doi.org/10.1073/pnas.1214574110 ![]() |
[65] |
Lopes L, Santos R, Pereira B, et al. (2013) Associations between gross motor coordination and academic achievement in elementary school children. Hum Mov Sci 32: 9-20. https://doi.org/10.1016/j.humov.2012.05.005 ![]() |
[66] |
Sibley BA, Etnier JL (2003) The relationship between physical activity and cognition in children: A meta-analysis. Pediatr Exercise Sci 15: 243-256. https://doi.org/10.1123/pes.15.3.243 ![]() |
[67] |
Tomporowski PD, McCullick B, Pendleton DM, et al. (2015) Exercise and children's cognition: The role of exercise characteristics and a place for metacognition. J Sport Health Sci 4: 47-55. https://doi.org/10.1016/j.jshs.2014.09.003 ![]() |
[68] |
Trudeau F, Shephard RJ (2008) Physical education, school physical activity, school sports and academic performance. Int J Behav Nutr Phys Act 5: 1-12. https://doi.org/10.1186/1479-5868-5-10 ![]() |
[69] |
Meijer A, Königs M, Vermeulen GT, et al. (2020) The effects of physical activity on brain structure and neurophysiological functioning in children: A systematic review and meta-analysis. Dev Cogn Neurosci 45: 100828. https://doi.org/10.1016/j.dcn.2020.100828 ![]() |
[70] |
Chaddock L, Erickson KI, Prakash RS, et al. (2010) Basal ganglia volume is associated with aerobic fitness in preadolescent children. Dev Neurosci 32: 249-256. https://doi.org/10.1159/000316648 ![]() |
[71] |
Chaddock L, Erickson KI, Prakash RS, et al. (2010) A neuroimaging investigation of the association between aerobic fitness, hippocampal volume, and memory performance in preadolescent children. Brain Res 1358: 172-183. https://doi.org/10.1016/j.brainres.2010.08.049 ![]() |
[72] | Stojiljković N, Mitić P, Sporiš G (2020) Can exercise make our children smarter?. Ann Kinesiol 10: 115-127. https://doi.org/10.35469/ak.2019.211 |
[73] |
Chaddock L, Erickson KI, Prakash RS, et al. (2012) A functional MRI investigation of the association between childhood aerobic fitness and neurocognitive control. Biol Psychol 89: 260-268. https://doi.org/10.1016/j.biopsycho.2011.10.017 ![]() |
[74] |
Pontifex MB, Raine LB, Johnson CR, et al. (2011) Cardiorespiratory fitness and the flexible modulation of cognitive control in preadolescent children. J Cogn Neurosci 23: 1332-1345. https://doi.org/10.1162/jocn.2010.21528 ![]() |
[75] |
Scudder MR, Federmeier KD, Raine LB, et al. (2014) The association between aerobic fitness and language processing in children: Implications for academic achievement. Brain Cogn 87: 140-152. https://doi.org/10.1016/j.bandc.2014.03.016 ![]() |
[76] |
Xue Y, Yang Y, Huang T (2019) Effects of chronic exercise interventions on executive function among children and adolescents: A systematic review with meta-analysis. Br J Sports Med 53: 1397-1404. https://doi.org/10.1136/bjsports-2018-099825 ![]() |
[77] |
Aguiar AS, Castro AA, Moreira EL, et al. (2011) Short bouts of mild-intensity physical exercise improve spatial learning and memory in aging rats: Involvement of hippocampal plasticity via AKT, CREB and BDNF signaling. Mech Ageing Dev 132: 560-567. https://doi.org/10.1016/j.mad.2011.09.005 ![]() |
[78] |
Raine LB, Khan NA, Drollette ES, et al. (2017) Obesity, visceral adipose tissue, and cognitive function in childhood. J Pediatr 187: 134-140.e3. https://doi.org/10.1016/j.jpeds.2017.05.023 ![]() |
[79] |
Pianta S, Lee JY, Tuazon JP, et al. (2019) A short bout of exercise prior to stroke improves functional outcomes by enhancing angiogenesis. Neuromol Med 21: 517-528. https://doi.org/10.1007/s12017-019-08533-x ![]() |
[80] |
Basso JC, Suzuki WA (2017) The effects of acute exercise on mood, cognition, neurophysiology, and neurochemical pathways: a review. Brain Plast 2: 127-152. https://doi.org/10.3233/bpl-160040 ![]() |
[81] |
Naylor AS, Persson AI, Eriksson PS, et al. (2005) Extended voluntary running inhibits exercise-induced adult hippocampal progenitor proliferation in the spontaneously hypertensive rat. J Neurophysiol 93: 2406-2414. https://doi.org/10.1152/jn.01085.2004 ![]() |
[82] |
Stein AM, Munive V, Fernandez AM, et al. (2017) Acute exercise does not modify brain activity and memory performance in APP/PS1 mice. PLoS One 12: e0178247. https://doi.org/10.1371/journal.pone.0178247 ![]() |
[83] |
Fernandes J, Soares JCK, do Amaral Baliego LGZ, et al. (2016) A single bout of resistance exercise improves memory consolidation and increases the expression of synaptic proteins in the hippocampus. Hippocampus 26: 1096-1103. https://doi.org/10.1002/hipo.22590 ![]() |
[84] |
da Silva de Vargas L, Neves BHS das, Roehrs R, et al. (2017) One-single physical exercise session after object recognition learning promotes memory persistence through hippocampal noradrenergic mechanisms. Behav Brain Res 329: 120-126. https://doi.org/10.1016/j.bbr.2017.04.050 ![]() |
[85] |
Rossi Daré L, Garcia A, Neves BH, et al. (2020) One physical exercise session promotes recognition learning in rats with cognitive deficits related to amyloid beta neurotoxicity. Brain Res 1744: 146918. https://doi.org/10.1016/j.brainres.2020.146918 ![]() |
[86] |
Nguemeni C, McDonald MW, Jeffers MS, et al. (2018) Short- and long-term exposure to low and high dose running produce differential effects on hippocampal neurogenesis. Neuroscience 369: 202-211. https://doi.org/10.1016/j.neuroscience.2017.11.026 ![]() |
[87] | Statton MA, Encarnacion M, Celnik P, et al. (2015) A single bout of moderate aerobic exercise improves motor skill acquisition. PLoS One 10: 1-13. https://doi.org/10.1371/journal.pone.0141393 |
[88] |
Smith AE, Goldsworthy MR, Garside T, et al. (2014) The influence of a single bout of aerobic exercise on short-interval intracortical excitability. Exp Brain Res 232: 1875-1882. https://doi.org/10.1007/s00221-014-3879-z ![]() |
[89] |
Giles GE, Brunyé TT, Eddy MD, et al. (2014) Acute exercise increases oxygenated and deoxygenated hemoglobin in the prefrontal cortex. Neuroreport 25: 1320-1325. https://doi.org/10.1097/WNR.0000000000000266 ![]() |
[90] |
Lulic T, El-Sayes J, Fassett HJ, et al. (2017) Physical activity levels determine exercise-induced changes in brain excitability. PLoS One 12: 1-18. https://doi.org/10.1371/journal.pone.0173672 ![]() |
[91] |
Suwabe K, Byun K, Hyodo K, et al. (2018) Rapid stimulation of human dentate gyrus function with acute mild exercise. Proc Natl Acad Sci 115: 10487-10492. https://doi.org/10.1073/pnas.1805668115 ![]() |
[92] |
Wagner G, Herbsleb M, de la Cruz F, et al. (2017) Changes in fMRI activation in anterior hippocampus and motor cortex during memory retrieval after an intense exercise intervention. Biol Psychol 124: 65-78. https://doi.org/10.1016/j.biopsycho.2017.01.003 ![]() |
[93] |
Chang YK, Labban JD, Gapin JI, et al. (2012) The effects of acute exercise on cognitive performance: A meta-analysis. Brain Res 1453: 87-101. https://doi.org/10.1016/j.brainres.2012.02.068 ![]() |
[94] |
Austin M, Loprinzi PD (2019) Acute exercise and mindfulness meditation on learning and memory: Randomized controlled intervention. Heal Promot Perspect 9: 314-318. https://doi.org/10.15171/hpp.2019.43 ![]() |
[95] |
Perini R, Bortoletto M, Capogrosso M, et al. (2016) Acute effects of aerobic exercise promote learning. Sci Rep 6: 25440. https://doi.org/10.1038/srep25440 ![]() |
[96] |
Chu CH, Alderman BL, Wei GX, et al. (2015) Effects of acute aerobic exercise on motor response inhibition: An ERP study using the stop-signal task. J Sport Health Sci 4: 73-81. https://doi.org/10.1016/j.jshs.2014.12.002 ![]() |
[97] |
Hsieh SS, Huang CJ, Wu CT, et al. (2018) Acute exercise facilitates the N450 inhibition marker and P3 attention marker during Stroop test in young and older adults. J Clin Med 7: 391. https://doi.org/10.3390/jcm7110391 ![]() |
[98] |
Samani A, Heath M (2018) Executive-related oculomotor control is improved following a 10-min single-bout of aerobic exercise: Evidence from the antisaccade task. Neuropsychologia 108: 73-81. https://doi.org/10.1016/j.neuropsychologia.2017.11.029 ![]() |
[99] |
Tsukamoto H, Suga T, Takenaka S, et al. (2016) Repeated high-intensity interval exercise shortens the positive effect on executive function during post-exercise recovery in healthy young males. Physiol Behav 160: 26-34. https://doi.org/10.1016/j.physbeh.2016.03.029 ![]() |
[100] |
Coles K, Tomporowski PD (2008) Effects of acute exercise on executive processing, short-term and long-term memory. J Sports Sci 26: 333-344. https://doi.org/10.1080/02640410701591417 ![]() |
[101] |
Hsieh SS, Chang YK, Hung TM, et al. (2016) The effects of acute resistance exercise on young and older males' working memory. Psychol Sport Exerc 22: 286-293. https://doi.org/10.1016/j.psychsport.2015.09.004 ![]() |
[102] |
Martins AQ, Kavussanu M, Willoughby A, et al. (2013) Moderate intensity exercise facilitates working memory. Psychol Sport Exerc 14: 323-328. https://doi.org/10.1016/j.psychsport.2012.11.010 ![]() |
[103] |
Weinberg L, Hasni A, Shinohara M, et al. (2014) A single bout of resistance exercise can enhance episodic memory performance. Acta Psychol 153: 13-19. https://doi.org/10.1016/j.actpsy.2014.06.011 ![]() |
[104] |
Winter B, Breitenstein C, Mooren FC, et al. (2007) High impact running improves learning. Neurobiol Learn Mem 87: 597-609. https://doi.org/10.1016/j.nlm.2006.11.003 ![]() |
[105] |
Dinoff A, Herrmann N, Swardfager W, et al. (2017) The effect of acute exercise on blood concentrations of brain-derived neurotrophic factor (BDNF) in healthy adults: A meta-analysis. Eur J Neurosci 46: 1635-1646. https://doi.org/10.1111/ejn.13603 ![]() |
[106] |
Moore D, Loprinzi PD (2020) Exercise influences episodic memory via changes in hippocampal neurocircuitry and long-term potentiation. Eur J Neurosci 54: 6960-6971. https://doi.org/10.1111/ejn.14728 ![]() |
[107] |
Ogoh S, Tsukamoto H, Hirasawa A, et al. (2014) The effect of changes in cerebral blood flow on cognitive function during exercise. Physiol Rep 2: 1-8. https://doi.org/10.14814/phy2.12163 ![]() |
[108] |
Steinborn MB, Huestegge L (2016) A walk down the lane gives wings to your brain. Restorative benefits of rest breaks on cognition and self-control. Appl Cogn Psychol 30: 795-805. https://doi.org/10.1002/acp.3255 ![]() |
[109] |
Chang YK, Liu S, Yu HH, et al. (2012) Effect of acute exercise on executive function in children with attention deficit hyperactivity disorder. Arch Clin Neuropsychol 27: 225-237. https://doi.org/10.1093/arclin/acr094 ![]() |
[110] | Nofuji Y, Suwa M, Sasaki H, et al. (2012) Different circulating brain-derived neurotrophic factor responses to acute exercise between physically active and sedentary subjects. J Sport Sci Med 11: 83-88. |
[111] |
Tomporowski PD (2003) Effects of acute bouts of exercise on cognition. Acta Psychol 112: 297-324. https://doi.org/10.1016/s0001-6918(02)00134-8 ![]() |
[112] |
Pontifex MB, McGowan AL, Chandler MC, et al. (2019) A primer on investigating the after effects of acute bouts of physical activity on cognition. Psychol Sport Exerc 40: 1-22. https://doi.org/10.1016/j.psychsport.2018.08.015 ![]() |
[113] |
Ferris LT, Williams JS, Shen CL (2007) The effect of acute exercise on serum brain-derived neurotrophic factor levels and cognitive function. Med Sci Sports Exerc 39: 728-734. https://doi.org/10.1249/mss.0b013e31802f04c7 ![]() |
[114] |
Schmitt A, Upadhyay N, Martin JA, et al. (2019) Modulation of distinct intrinsic resting state brain networks by acute exercise bouts of differing intensity. Brain Plast 5: 39-55. https://doi.org/10.3233/bpl-190081 ![]() |
[115] |
Mehren A, Luque CD, Brandes M, et al. (2019) Intensity-dependent effects of acute exercise on executive function. Neural Plast 2019: 1-17. https://doi.org/10.1155/2019/8608317 ![]() |
[116] |
Alves CRR, Tessaro VH, Teixeira LAC, et al. (2014) Influence of acute high-intensity aerobic interval exercise bout on selective attention and short-term memory tasks. Percept Mot Skills 118: 63-72. https://doi.org/10.2466/22.06.PMS.118k10w4 ![]() |
[117] |
Du Rietz E, Barker AR, Michelini G, et al. (2019) Beneficial effects of acute high-intensity exercise on electrophysiological indices of attention processes in young adult men. Behav Brain Res 359: 474-484. https://doi.org/10.1016/j.bbr.2018.11.024 ![]() |
[118] |
Roig M, Skriver K, Lundbye-Jensen J, et al. (2012) A single bout of exercise improves motor memory. PLoS One 7: e44594. https://doi.org/10.1371/journal.pone.0044594 ![]() |
[119] | Thomas R, Johnsen LK, Geertsen SS, et al. (2016) Acute exercise and motor memory consolidation: The role of exercise intensity. PLoS One 11: 1-16. https://doi.org/10.1371/journal.pone.0159589 |
[120] |
Grego F, Vallier JM, Collardeau M, et al. (2005) Influence of exercise duration and hydration status on cognitive function during prolonged cycling exercise. Int J Sports Med 26: 27-33. https://doi.org/10.1055/s-2004-817915 ![]() |
[121] |
Ai JY, Chen FT, Hsieh SS, et al. (2021) The effect of acute high-intensity interval training on executive function: A systematic review. Int J Environ Res Public Health 18: 3593. https://doi.org/10.3390/ijerph18073593 ![]() |
[122] |
Loprinzi PD, Roig M, Etnier JL, et al. (2021) Acute and chronic exercise effects on human memory: What we know and where to go from here. J Clin Med 10: 4812. https://doi.org/10.3390/jcm10214812 ![]() |
[123] |
Benzing V, Heinks T, Eggenberger N, et al. (2016) Acute cognitively engaging exergame-based physical activity enhances executive functions in adolescents. PLoS One 11: 1-15. https://doi.org/10.1371/journal.pone.0167501 ![]() |
[124] |
Pesce C, Crova C, Cereatti L, et al. (2009) Physical activity and mental performance in preadolescents: Effects of acute exercise on free-recall memory. Ment Health Phys Act 2: 16-22. https://doi.org/10.1016/j.mhpa.2009.02.001 ![]() |
[125] |
Berman MG, Jonides J, Kaplan S (2008) The cognitive benefits of interacting with nature. Psychol Sci 19: 1207-1212. https://doi.org/10.1111/j.1467-9280.2008.02225.x ![]() |
[126] |
O'Leary KC, Pontifex MB, Scudder MR, et al. (2011) The effects of single bouts of aerobic exercise, exergaming, and videogame play on cognitive control. Clin Neurophysiol 122: 1518-1525. https://doi.org/10.1016/j.clinph.2011.01.049 ![]() |
[127] |
Thomas R, Flindtgaard M, Skriver K, et al. (2017) Acute exercise and motor memory consolidation: Does exercise type play a role?. Scand J Med Sci Sport 27: 1523-1532. https://doi.org/10.1111/sms.12791 ![]() |
[128] |
Labelle V, Bosquet L, Mekary S, et al. (2013) Decline in executive control during acute bouts of exercise as a function of exercise intensity and fitness level. Brain Cogn 81: 10-17. https://doi.org/10.1016/j.bandc.2012.10.001 ![]() |
[129] |
Tsai CL, Chen FC, Pan CY, et al. (2014) Impact of acute aerobic exercise and cardiorespiratory fitness on visuospatial attention performance and serum BDNF levels. Psychoneuroendocrinology 41: 121-131. https://doi.org/10.1016/j.psyneuen.2013.12.014 ![]() |
[130] |
Colcombe SJ, Kramer AF, Erickson KI, et al. (2004) Cardiovascular fitness, cortical plasticity, and aging. Proc Natl Acad Sci USA 101: 3316-3321. https://doi.org/10.1073/pnas.0400266101 ![]() |
[131] |
Holzschneider K, Wolbers T, Röder B, et al. (2012) Cardiovascular fitness modulates brain activation associated with spatial learning. Neuroimage 59: 3003-3014. https://doi.org/10.1016/j.neuroimage.2011.10.021 ![]() |
[132] |
Dupuy O, Bosquet L, Fraser SA, et al. (2018) Higher cardiovascular fitness level is associated to better cognitive dual-task performance in Master Athletes: Mediation by cardiac autonomic control. Brain Cogn 125: 127-134. https://doi.org/10.1016/j.bandc.2018.06.003 ![]() |
[133] |
Schwarb H, Johnson CL, Daugherty AM, et al. (2017) Aerobic fitness, hippocampal viscoelasticity, and relational memory performance. Neuroimage 153: 179-188. https://doi.org/10.1016/j.neuroimage.2017.03.061 ![]() |
[134] |
Hüttermann S, Memmert D (2014) Does the inverted-U function disappear in expert athletes? An analysis of the attentional behavior under physical exercise of athletes and non-athletes. Physiol Behav 131: 87-92. https://doi.org/10.1016/j.physbeh.2014.04.020 ![]() |
[135] |
Hwang J, Castelli DM, Gonzalez-Lima F (2017) The positive cognitive impact of aerobic fitness is associated with peripheral inflammatory and brain-derived neurotrophic biomarkers in young adults. Physiol Behav 179: 75-89. https://doi.org/10.1016/j.physbeh.2017.05.011 ![]() |
[136] |
Hopkins ME, Davis FC, Vantieghem MR, et al. (2012) Differential effects of acute and regular physical exercise on cognition and affect. Neuroscience 215: 59-68. https://doi.org/10.1016/j.neuroscience.2012.04.056 ![]() |
[137] |
Hübner L, Voelcker-Rehage C (2017) Does physical activity benefit motor performance and learning of upper extremity tasks in older adults?—A systematic review. Eur Rev Aging Phys Act 14: 1-19. https://doi.org/10.1186/s11556-017-0181-7 ![]() |
[138] |
Roig M, Thomas R, Mang CS, et al. (2016) Time-dependent effects of cardiovascular exercise on memory. Exerc Sport Sci Rev 44: 81-88. https://doi.org/10.1249/JES.0000000000000078 ![]() |
[139] |
Lind RR, Beck MM, Wikman J, et al. (2019) Acute high-intensity football games can improve children's inhibitory control and neurophysiological measures of attention. Scand J Med Sci Sport 29: 1546-1562. https://doi.org/10.1111/sms.13485 ![]() |
[140] |
Thomas R, Beck MM, Lind RR, et al. (2016) Acute exercise and motor memory consolidation: The role of exercise timing. Neural Plast 2016: 1-11. https://doi.org/10.1155/2016/6205452 ![]() |
[141] |
van Dongen EV, Kersten IHP, Wagner IC, et al. (2016) Physical exercise performed four hours after learning improves memory retention and increases hippocampal pattern similarity during retrieval. Curr Biol 26: 1722-1727. https://doi.org/10.1016/j.cub.2016.04.071 ![]() |
[142] |
Loprinzi PD, Blough J, Crawford L, et al. (2019) The temporal effects of acute exercise on episodic memory function: Systematic review with meta-analysis. Brain Sci 9: 87. https://doi.org/10.3390/brainsci9040087 ![]() |
[143] |
Zhang B, Liu Y, Zhao M, et al. (2020) Differential effects of acute physical activity on executive function in preschoolers with high and low habitual physical activity levels. Ment Health Phys Act 18: 100326. https://doi.org/10.1016/j.mhpa.2020.100326 ![]() |
[144] |
Oberste M, Javelle F, Sharma S, et al. (2019) Effects and moderators of acute aerobic exercise on subsequent interference control: a systematic review and meta-analysis. Front Psychol 10: 2616. https://doi.org/10.3389/fpsyg.2019.02616 ![]() |
[145] |
Budde H, Voelcker-Rehage C, Pietraßyk-Kendziorra S, et al. (2008) Acute coordinative exercise improves attentional performance in adolescents. Neurosci Lett 441: 219-223. https://doi.org/10.1016/j.neulet.2008.06.024 ![]() |
[146] |
Ellemberg D, St-Louis-Deschênes M (2010) The effect of acute physical exercise on cognitive function during development. Psychol Sport Exerc 11: 122-126. https://doi.org/10.1016/j.psychsport.2009.09.006 ![]() |
[147] |
Chen AG, Yan J, Yin HC, et al. (2014) Effects of acute aerobic exercise on multiple aspects of executive function in preadolescent children. Psychol Sport Exerc 15: 627-636. https://doi.org/10.1016/j.psychsport.2014.06.004 ![]() |
[148] |
Ludyga S, Gerber M, Brand S, et al. (2016) Acute effects of moderate aerobic exercise on specific aspects of executive function in different age and fitness groups: A meta-analysis. Psychophysiology 53: 1611-1626. https://doi.org/10.1111/psyp.12736 ![]() |
[149] |
Etnier J, Labban JD, Piepmeier A, et al. (2014) Effects of an acute bout of exercise on memory in 6th grade children. Pediatr Exerc Sci 26: 250-258. https://doi.org/10.1123/pes.2013-0141 ![]() |
[150] |
Pesce C, Conzelmann A, Jäger K, et al. (2017) Disentangling the relationship between children's motor ability, executive function and academic achievement. PLoS One 12: e0182845. https://doi.org/10.1371/journal.pone.0182845 ![]() |
[151] |
Williams RA, Hatch L, Cooper SB (2019) A review of factors affecting the acute exercise-cognition relationship in children and adolescents. OBM Integr Complement Med 4: 1. https://doi.org/10.21926/OBM.ICM.1903049 ![]() |
[152] |
Angulo-Barroso R, Ferrer-Uris B, Busquets A (2019) Enhancing children's motor memory retention through acute intense exercise: Effects of different exercise durations. Front Psychol 10: 1-9. https://doi.org/10.3389/fpsyg.2019.02000 ![]() |
[153] |
Budde H, Voelcker-Rehage C, Pietraßyk-Kendziorra S, et al. (2008) Acute coordinative exercise improves attentional performance in adolescents. Neurosci Lett 441: 219-223. https://doi.org/10.1016/j.neulet.2008.06.024 ![]() |
[154] |
Jäger K, Schmidt M, Conzelmann A, et al. (2015) The effects of qualitatively different acute physical activity interventions in real-world settings on executive functions in preadolescent children. Ment Health Phys Act 9: 1-9. https://doi.org/10.1016/j.mhpa.2015.05.002 ![]() |
[155] |
Hillman CH, Pontifex MB, Raine LB, et al. (2009) The effect of acute treadmill walking on cognitive control and academic achievement in preadolescent children. Neuroscience 159: 1044-1054. https://doi.org/10.1016/j.neuroscience.2009.01.057 ![]() |
[156] |
Stroth S, Kubesch S, Dieterle K, et al. (2009) Physical fitness, but not acute exercise modulates event-related potential indices for executive control in healthy adolescents. Brain Res 1269: 114-124. https://doi.org/10.1016/j.brainres.2009.02.073 ![]() |
[157] |
Tomporowski PD, Davis CL, Lambourne K, et al. (2008) Task switching in overweight children: Effects of acute exercise and age. J Sport Exerc Psychol 30: 497-511. https://doi.org/10.1123/jsep.30.5.497 ![]() |
[158] |
Maltais DB, Gane C, Dufour SK, et al. (2016) Acute physical exercise affects cognitive functioning in children With cerebral palsy. Pediatr Exerc Sci 28: 304-311. https://doi.org/10.1123/pes.2015-0110 ![]() |
[159] | Villa-González R, Villalba-Heredia L, Crespo I, et al. (2020) A systematic review of acute exercise as a coadjuvant treatment of ADHD in young people. Psicothema 32: 67-74. https://doi.org/10.7334/psicothema2019.211 |
[160] |
Bremer E, Graham JD, Heisz JJ, et al. (2020) Effect of acute exercise on prefrontal oxygenation and inhibitory control among male children with autism spectrum disorder: an exploratory study. Front Behav Neurosci 14: 1-10. https://doi.org/10.3389/fnbeh.2020.00084 ![]() |
[161] |
Metcalfe AWS, MacIntosh BJ, Scavone A, et al. (2016) Effects of acute aerobic exercise on neural correlates of attention and inhibition in adolescents with bipolar disorder. Transl Psychiatry 6: e814. https://doi.org/10.1038/tp.2016.85 ![]() |
[162] |
Ng QX, Ho CYX, Chan HW, et al. (2017) Managing childhood and adolescent attention-deficit/hyperactivity disorder (ADHD) with exercise: A systematic review. Complement Ther Med 34: 123-128. https://doi.org/10.1016/j.ctim.2017.08.018 ![]() |
[163] |
Suarez-Manzano S, Ruiz-Ariza A, De La Torre-Cruz M, et al. (2018) Acute and chronic effect of physical activity on cognition and behaviour in young people with ADHD: A systematic review of intervention studies. Res Dev Disabil 77: 12-23. https://doi.org/10.1016/j.ridd.2018.03.015 ![]() |
[164] |
Smith AL, Hoza B, Linnea K, et al. (2013) Pilot physical activity intervention reduces severity of ADHD symptoms in young children. J Atten Disord 17: 70-82. https://doi.org/10.1177/1087054711417395 ![]() |
[165] |
Vysniauske R, Verburgh L, Oosterlaan J, et al. (2016) The effects of physical exercise on functional outcomes in the treatment of ADHD: a meta-analysis. J Atten Disord 24: 644-654. https://doi.org/10.1177/1087054715627489 ![]() |
[166] |
Chandrasekaran B, Pesola AJ, Rao CR, et al. (2021) Does breaking up prolonged sitting improve cognitive functions in sedentary adults? A mapping review and hypothesis formulation on the potential physiological mechanisms. BMC Musculoskelet Disord 22: 1-16. https://doi.org/10.1186/s12891-021-04136-5 ![]() |
[167] |
Edington DW, Schultz AB, Pitts JS, et al. (2015) The future of health promotion in the 21st century: a focus on the working population. Am J Lifestyle Med 10: 242-252. https://doi.org/10.1177/1559827615605789 ![]() |
[168] |
Taylor WC (2005) Transforming work breaks to promote health. Am J Prev Med 29: 461-465. https://doi.org/10.1016/j.amepre.2005.08.040 ![]() |
[169] |
Taylor WC, King KE, Shegog R, et al. (2013) Booster breaks in the workplace: participants' perspectives on health-promoting work breaks. Health Educ Res 28: 414-425. https://doi.org/10.1093/her/cyt001 ![]() |
[170] |
Wollseiffen P, Ghadiri A, Scholz A, et al. (2015) Short bouts of intensive exercise during the workday have a positive effect on neuro-cognitive performance. Stress Health 32: 514-523. https://doi.org/10.1002/smi.2654 ![]() |
[171] |
Mahar MT, Murphy SK, Rowe DA, et al. (2006) Effects of a classroom-based program on physical activity and on-task behavior. Med Sci Sports Exercise 38: 2086-2094. https://doi.org/10.1249/01.mss.0000235359.16685.a3 ![]() |
[172] |
Mahar MT (2011) Impact of short bouts of physical activity on attention-to-task in elementary school children. Prev Med 52: S60-S64. https://doi.org/10.1016/j.ypmed.2011.01.026 ![]() |
[173] | Schmidt M, Benzing V, Kamer M (2016) Classroom-based physical activity breaks and children's attention: Cognitive engagement works!. Front Psychol 7: 1474. https://doi.org/10.3389/fpsyg.2016.01474 |
[174] |
Engelen L, Chau J, Young S, et al. (2018) Is activity-based working impacting health, work performance and perceptions? A systematic review. Build Res Inf 47: 468-479. https://doi.org/10.1080/09613218.2018.1440958 ![]() |
[175] |
Arundell L, Sudholz B, Teychenne M, et al. (2018) The impact of activity based working (ABW) on workplace activity, eating behaviours, productivity, and satisfaction. Int J Environ Res Public Health 15: 1005. https://doi.org/10.3390/ijerph15051005 ![]() |
[176] |
Proper KI, Staal BJ, Hildebrandt VH, et al. (2002) Effectiveness of physical activity programs at worksites with respect to work-related outcomes. Scand J Work Environ Health 28: 75-84. https://doi.org/10.5271/sjweh.651 ![]() |
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