Citation: Yannick Lutz, Rosa Daschner, Lorena Krames, Axel Loewe, Giorgio Cattaneo, Stephan Meckel, Olaf Dössel. Modeling selective therapeutic hypothermia in case of acute ischemic stroke using a 1D hemodynamics model and a simplified brain geometry[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1147-1167. doi: 10.3934/mbe.2020060
[1] | Ghulam Farid, Hafsa Yasmeen, Hijaz Ahmad, Chahn Yong Jung . Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly (α,m)-convex functions. AIMS Mathematics, 2021, 6(10): 11403-11424. doi: 10.3934/math.2021661 |
[2] | Yu-Pei Lv, Ghulam Farid, Hafsa Yasmeen, Waqas Nazeer, Chahn Yong Jung . Generalization of some fractional versions of Hadamard inequalities via exponentially (α,h−m)-convex functions. AIMS Mathematics, 2021, 6(8): 8978-8999. doi: 10.3934/math.2021521 |
[3] | Shuang-Shuang Zhou, Ghulam Farid, Chahn Yong Jung . Convexity with respect to strictly monotone function and Riemann-Liouville fractional Fejér-Hadamard inequalities. AIMS Mathematics, 2021, 6(7): 6975-6985. doi: 10.3934/math.2021409 |
[4] | Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386 |
[5] | Moquddsa Zahra, Muhammad Ashraf, Ghulam Farid, Kamsing Nonlaopon . Inequalities for unified integral operators of generalized refined convex functions. AIMS Mathematics, 2022, 7(4): 6218-6233. doi: 10.3934/math.2022346 |
[6] | Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak . Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546 |
[7] | Jamshed Nasir, Shahid Qaisar, Saad Ihsan Butt, Hassen Aydi, Manuel De la Sen . Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications. AIMS Mathematics, 2022, 7(3): 3418-3439. doi: 10.3934/math.2022190 |
[8] | Ghulam Farid, Maja Andrić, Maryam Saddiqa, Josip Pečarić, Chahn Yong Jung . Refinement and corrigendum of bounds of fractional integral operators containing Mittag-Leffler functions. AIMS Mathematics, 2020, 5(6): 7332-7349. doi: 10.3934/math.2020469 |
[9] | Manar A. Alqudah, Artion Kashuri, Pshtiwan Othman Mohammed, Muhammad Raees, Thabet Abdeljawad, Matloob Anwar, Y. S. Hamed . On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space. AIMS Mathematics, 2021, 6(5): 4638-4663. doi: 10.3934/math.2021273 |
[10] | Atiq Ur Rehman, Ghulam Farid, Sidra Bibi, Chahn Yong Jung, Shin Min Kang . k-fractional integral inequalities of Hadamard type for exponentially (s,m)-convex functions. AIMS Mathematics, 2021, 6(1): 882-892. doi: 10.3934/math.2021052 |
A function plays very vital role in studying real analysis, numerical analysis, functional analysis and statistical analysis etc. Properties of a function like continuity, differentiability increase its importance in the subjects of mathematical analysis, economics, differential equations etc. A convex function defined about a century ago has fascinating geometric and analytical properties, due to which it is studied frequently in the subjects of mathematical analysis and optimization theory. Its extended form is the well-known Jensen inequality which in particular generates several classical inequalities. In the development of the theory of mathematical inequalities its role is significant. Description of a convex function in different convenient forms motivates the researchers to extend its concept in the form of new definitions and notions. Especially, to generalize and refine the classical inequalities for convex functions, it can be found in the literature that researchers have defined many new classes of functions.
In the theory of inequalities the class of convex functions has been considered at very large scale in past few decades, see [1,2,3,4,5] and references therein.
A function f defined on an interval I is called convex if the following inequality is satisfied [6]:
f(tx+(1−t)y)≤tf(x)+(1−t)f(y),t∈[0,1],x,y∈ I. | (1.1) |
A function f is said to be convex with respect to a strictly monotone function g if f∘g−1 is a convex function, see [7,Definition 1.19,p.7]. An analytic representation (1.1) of convex function provides the motivation to define new classes of functions. By including some convenient parameters and functions, various numerous generalizations, extensions and refinements have been introduced. For example in [8,9,10,11,12], authors have defined m-convex, h-convex, s-convex, Godunova-Levin convex, Godunova-Levin s-convex, (α,m)-convex, (h−m)-convex, (s,m)-convex and P-convex functions. All such functions can be reproduced from the following class of functions called (α,h−m)-convex functions.
Let J⊆R be an interval containing (0,1) and let h:J→R be a non-negative function, h≢0. A function f:[0,b]→R is said to be (α,h−m)-convex, if f is non-negative and for all x,y∈[0,b], (α,m)∈[0,1]2, t∈(0,1), we have
f(tx+m(1−t)y)≤h(tα)f(x)+mh(1−tα)f(y), | (1.2) |
see [4] for more details.
In [13], authors defined a new notion of functions as follows:
Let h:J⊂R→R be a function with h≢0. A function f:I→R is said to be (α,h−m)-convex with respect to a strictly monotone function g:K⊂I→R, if f,h≥0 and for each x,y∈I, we have
f∘g−1(tx+m(1−t)y)≤h(tα)f∘g−1(x)+mh(1−tα)f∘g−1(y), | (1.3) |
where (α,m)∈[0,1]2 and t∈(0,1). This idea is extended and in [14] refined (α,h−m)-convex function has been introduced which provide refinements of various kinds of convex functions which have been studied in [15,16,17] can be reproduced.
Let h:J⊆R→R be a function with J containing (0,1). A function f:I→R is called refined (α,h−m)-convex function, if f,h≥0 and for each x,y∈I, we have
f(tx+m(1−t)y)≤h(tα)h(1−tα)(f(x)+mf(y)), | (1.4) |
where (α,m)∈(0,1]2 and t∈(0,1), see [14] for more details.
The aim of this work is to introduce refined (α,h−m)-convex function with respect to strictly monotone function. By applying this new definition we will establish generalized Riemann-Liouville fractional integral inequalities. These inequalities will also provide refinements of Riemann-Liouville fractional integral inequalities for specific functions involved in Definition 2.1.
Fractional calculus operators of differentiation and integration are key factors in generalizing classical concepts of calculus. The role of derivatives and their applications in diverse fields of science and engineering is remarkable. Fractional calculus operators generalize all the classical phenomena related to usual derivative and integration, see [18] for more details.
The Riemann-Liouville fractional and derivative integral operators are the most classical well-known integrals and derivatives of fractional order defined as follows:
Definition 1.1. [18] Let f∈L1[u,v]. Then Riemann-Liouville fractional integrals of order β of a function f are given as follows:
Iβu+f(x)=1Γ(β)∫xu(x−t)β−1f(t)dt,x>u, | (1.5) |
Iβv−f(y)=1Γ(β)∫vy(t−y)β−1f(t)dt,y<v, | (1.6) |
where
Γ(β)=∫∞0tβ−1e−tdt, |
and ℜ(β)>0.
Definition 1.2. [19] Let f∈L1[u,v]. Then k-fractional Riemann-Liouville integrals of order β of a function f are given as follows:
kIβu+f(x)=1kΓk(β)∫xu(x−t)βk−1f(t)dt,x>u, | (1.7) |
kIβv−f(y)=1kΓk(β)∫vy(t−y)βk−1f(t)dt,y<v, | (1.8) |
where
Γk(β)=∫∞0tβ−1e−tkkdt, |
and ℜ(β),k>0. Using Γk(β)=kβk−1Γ(βk) in (1.5) and (1.6), one can get
k−βkIβu+f(x)=kIβu+f(x) |
and
k−βkIβv−f(y)=kIβv−f(y). |
The Hadamard inequality is of immense importance in the theory of inequalities and a lot of work has been published about this inequality in the past few decades, see [20,21,22,23]. The Hadamard inequality is stated as follows:
Let f be a convex function on [u,v] with u<v. Then, the following inequality holds:
f(u+v2)≤1v−u∫vuf(x)dx≤f(u)+f(v)2. | (1.9) |
A number of numerous generalizations and refinements of the inequality (1.9) have been published in recent years, see [11,14,24]. Keeping in view these results we aim to find some new refinements of existing inequalities by using refined (α,h−m)-convex functions with respect to a strictly monotone function. These results are also applicable for k-fractional version of Riemann-Liouville integrals which are also given in this paper.
The rest of the paper is organized as follows: In Section 2, we define a new class of convex functions, namely refined (α,h−m)-convex function with respect to a strictly monotone function and establish Hadamard type inequalities by using this class of functions. The k-analogue versions of these results are given in Section 3. Moreover, the results of this paper are connected with already published inequalities. In Section 4, we give concluding remarks.
Definition 2.1. Let h:J⊆R→R be a function with (0,1)⊆J. Also, let K,I are intervals in R such that K⊂I. A function f:I→R is called refined (α,h−m)-convex function with respect to strictly monotone function g:K→R, range(g)⊂I if f,h≥0 and for each x,y∈I, we have
f∘g−1(tx+m(1−t)y)≤h(tα)h(1−tα)(f∘g−1(x)+mf∘g−1(y)), | (2.1) |
where (α,m)∈(0,1]2 and t∈(0,1).
By comparing inequalities (1.3) and (2.1), it can be realized that if 0<h(t)<1, then (2.1) reduces to the following refinement of (α,h−m)-convex function with respect to strictly monotone function g:
f∘g−1(tx+m(1−t)y)≤h(tα)h(1−tα)(f∘g−1(x)+mf∘g−1(y))=h(tα)h(1−tα)f∘g−1(x)+mh(tα)h(1−tα)f∘g−1(y)≤h(tα)f∘g−1(x)+mh(1−tα)f∘g−1(y). | (2.2) |
Next, we give some examples as follows:
Example 2.1. The function x2+1 is (1,Id−1)-convex as well as refined (1,Id−1)-convex with respect to x on (0,1). But on (1,∞), it is (1,Id−1)-convex but not refined (1,Id−1)-convex with respect to x.
Example 2.2. Let g(x)=xp for x∈(0,∞) and p∈R∖{0}. Then g is strictly monotone function (strictly increasing for p>0 and strictly decreasing for p<0) and g−1(x)=x1p. By using Definition 2.1, we have
f((tx+m(1−t)y))1p)≤h(tα)h(1−tα)(f(x1p)+mf(y1p)). | (2.3) |
By replacing x with xp and y with yp, we get
f((txp+m(1−t)yp))1p)≤h(tα)h(1−tα)(f(x)+mf(y)). | (2.4) |
Inequality (2.4) gives the definition of refined (α,h−m)−p-convex function given in [25]. Whereas for p=−1, the inequality (2.3) reduces to the following inequality:
f(1(tx+m(1−t)y))≤h(tα)h(1−tα)(f(1x)+mf(1y)). | (2.5) |
By replacing x with 1x and y with 1y, we get
f(xy(ty+m(1−t)x))≤h(tα)h(1−tα)(f(x)+mf(y)), | (2.6) |
that is f∘g−1 is refined harmonically (α,h−m)-convex function defined in [25].
For h(t)=t in (2.1), we get definition for refined (α,m)-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.2. A function f:I→R is called refined (α,m)-convex function with respect to a strictly monotone function g:K⊂I→R, if for each x,y∈I, we have
f∘g−1(tx+m(1−t)y)≤tα(1−tα)(f∘g−1(x)+mf∘g−1(y)), | (2.7) |
where (α,m)∈(0,1]2 and t∈(0,1).
By taking h(t)=t and m=1, (2.1) gives the definition for refined α-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.3. A function f:I→R is called refined α-convex function with respect to a strictly monotone function g:K⊂I→R, if for each x,y∈I, we have
f∘g−1(tx+(1−t)y)≤tα(1−tα)(f∘g−1(x)+f∘g−1(y)), | (2.8) |
where m∈(0,1] and t∈(0,1).
By taking α=1, (2.1) gives the definition of refined (h−m)-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.4. Let h:J⊆R→R be a function with J containing (0,1). A function f:I→R is called refined (h−m)-convex function with respect to a strictly monotone function g:K⊂I→R, if f,h≥0 and for each x,y∈I, one have the inequality
f∘g−1(tx+m(1−t)y)≤h(t)h(1−t)(f∘g−1(x)+mf∘g−1(y)), | (2.9) |
where m∈[0,1] and t∈(0,1).
By taking α=1=m, (2.1) gives the definition of refined h-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.5. Let h:J⊆R→R be a function with J containing (0,1). A function f:I→R is called refined h-convex function with respect to a strictly monotone function g:K⊂I→R, if f,h≥0 and for each x,y∈I, one have the inequality
f∘g−1(tx+(1−t)y)≤h(t)h(1−t)(f∘g−1(x)+f∘g−1(y)), | (2.10) |
where t∈(0,1).
By taking α=1 and h(t)=ts, (2.1) gives the definition of refined (s,m)-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.6. A function f:I→R is called refined (s,m)-convex function with respect to a strictly monotone function g:K⊂I→R, if for each x,y∈I, we have
f∘g−1(tx+m(1−t)y)≤ts(1−t)s(f∘g−1(x)+mf∘g−1(y)), | (2.11) |
where (s,m)∈(0,1]2 and t∈(0,1).
By taking α=1 and h(t)=t, (2.1) gives the definition of refined m-convex function with respect to a strictly monotone function g stated as follows:
Definition 2.7. A function f:I→R is called refined m-convex function with respect to a strictly monotone function g:K⊂I→R, if for each x,y∈I, we have
f∘g−1(tx+m(1−t)y)≤t(1−t)(f∘g−1(x)+mf∘g−1(y)), | (2.12) |
where m∈(0,1] and t∈(0,1).
Now, we give the following Hadamard type inequalities for Riemann-Liouville fractional integral operators via refined (α,h−m)-convex function with respect to a strictly monotone function g.
Theorem 2.1. Let f:[u,mv]→R with 0≤u<mv and f∈L1[u,v]. Also, suppose that f≥0 is refined (α,h−m)-convex function with respect to a strictly monotone function g. Then for (α,m)∈(0,1]2, the following fractional integral inequality holds:
f∘g−1(u+mv2)h(12α)h(2α−12α)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα)h(1−tα)tβ−1dt, | (2.13) |
with β>0.
Proof. Since f is refined (α,h−m)-convex function with respect to a strictly monotone function g, by using (2.1), one can have the following inequality:
f∘g−1(x+my2)≤h(12α)h(2α−12α)[f∘g−1(x)+mf∘g−1(y)]. | (2.14) |
By setting x=tu+m(1−t)v and y=um(1−t)+vt in (2.14), the following inequality can be yielded:
f∘g−1(u+mv2)≤h(12α)h(2α−12α)(f∘g−1(tu+m(1−t)v)+mf∘g−1(um(1−t)+vt)). |
Now, by integrating the above inequality over the interval [0,1] after multiplying with tβ−1, one can have the following inequality:
f∘g−1(u+mv2)βh(12α)h(2α−12α)≤∫10f∘g−1(tu+m(1−t)v)tβ−1dt+m∫10f∘g−1(um(1−t)+vt)tβ−1dt. | (2.15) |
On the right hand side by applying definition of refined (α,h−m)-convexity with respect to a strictly monotone function g, one can obtain the following inequality:
![]() |
(2.16) |
Making change of variables, then by applying Definition 1.1 and multiplying by β, one can easily obtain the inequality required in (2.13). The proof is completed.
Remark 2.1. (i) For g(x)=x, (2.13) reduces to [14,Theorem 1].
(ii) For m=α=1 and h(t)=t, (2.13) reduces to [24,Theorem 3.1].
(iii) For β=1 along with the conditions of (ii), (2.13) reduces to [24,Theorem 2.1].
(vi) For h(t)=t along with the condition of (i), (2.13) reduces to [14,Corollary 1].
(vii) For α=1 along with the condition of (i), (2.13) reduces to [14,Corollary 2].
(viii) For α=1 and h(t)=ts along with the condition of (i), (2.13) reduces to [14,Corollary 4].
(ix) For α=1 and h(t)=t along with the condition of (i), (2.13) reduces to [14,Corollary 5].
(x) For m=1 in the result of (x), (2.13) reduces to [14,Corollary 6].
By imposing an additional condition, we get the following result which shows the extension of Theorem 2.1.
Theorem 2.2. Along with} the assumptions of Theorem 2.1, if h is bounded above by 1√2, then the following inequality holds:
2f∘g−1(u+mv2)≤1h(12α)h(2α−12α)f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα)h(1−tα)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.17) |
Proof. By using the condition h(t)≤1√2, one can have h(tα)h(1−tα)≤12. Integrating over [0,1] after multiplying with tβ−1, we get the following inequality:
∫10h(tα)h(1−tα)tβ−1dt≤12β. | (2.18) |
Since f is refined (α,h−m)-convex function with respect to a strictly monotone function g, by using (2.1), one can have the following inequality:
1h(12α)h(2α−12α)f∘g−1(u+mv2)≥2f∘g−1(u+mv2). | (2.19) |
From inequalities (2.13), (2.18) and (2.19), one can get (2.17).
Remark 2.2. (i) For g(x)=x, (2.17) reduces to [14,Theorem 2].
(ii) For g(x)=xp, (2.17) reduces to [25,Theorem 2].
(iii) For α=1 along with the condition of (i), (2.17) reduces to [14,Corollary 3].
(v) For α=1 along with the condition of (ii), (2.17) reduces to [25,Corollary 3].
Corollary 2.1. Using h(t)=t, (2.13) gives the fractional integral inequality for refined (α,m)-convex function with respect to a strictly monotone function g in the following form:
22α2α−1f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤αβ(β+α)(β+2α)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.20) |
Corollary 2.2. Applying α=1, (2.13) gives the fractional integral inequality for refined (h−m)-convex function with respect to strictly monotone function g in the following form:
1h2(12)f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(t)h(1−t)tβ−1dt. | (2.21) |
Corollary 2.3. Applying α=1 and h(t)=ts, (2.13) gives the fractional integral inequality for refined (s,m)-convex function with respect to a strictly monotone function g in the following form:
22sf∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]B(1+s,s+β), | (2.22) |
where B(1+s,s+β) is the beta function.
Corollary 2.4. Applying α=1 and h(t)=t, (2.13) gives the inequality for refined m-convex function with respect to a strictly monotone g in the following form:
4f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β(β+1)(β+2)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.23) |
Corollary 2.5. Applying α=1, (2.17) gives the fractional integral inequality for refined (h−m)-convex function with respect to a strictly monotone function g in the following form:
2f∘g−1(u+mv2)≤1h2(12)f∘g−1(u+mv2)≤Γ(β+1)(mv−u)β[Iβu+f∘g−1(mv)+mβ+1Iβv−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(t)h(1−t)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.24) |
Theorem 2.3. Under the assumption of Theorem 2.1, the following fractional integral inequality holds:
![]() |
(2.25) |
with β>0.
Proof. We use the inequality (2.14) to prove this theorem. By setting x=tu2+m(2−t2)v and y=um(2−t2)+tv2 in (2.14), one can get the following inequality:
![]() |
By integrating the above inequality over the interval [0,1] after multiplying with tβ−1, one can have the following inequality:
![]() |
(2.26) |
On the right hand side by applying definition of refined (α,h−m)-convexity with respect to a strictly monotone function g, one can obtain the following inequality:
![]() |
(2.27) |
Making change of variables, then by applying Definition 1.1 and multiplying by β, one can easily obtain the inequality required in (2.25).
Remark 2.3. (i) For g(x)=x, (2.25) reduces to [14,Theorem 3].
(ii) For β=m=α=1 and h(t)=t, (2.25) reduces to [24,Theorem 2.1].
(iii) For h(t)=t along with the condition of (i), (2.25) reduces to [14,Corollary 7].
(iv) For α=1 along with the condition of (i), (2.25) reduces to [14,Corollary 8].
(v) For α=1 and h(t)=ts along with the condition of (i), (2.25) reduces to [14,Corollary 10].
(vi) For α=1 and h(t)=t along with the condition of (i), (2.25) reduces to [14,Corollary 11].
(vii) For m=1 in the result of (i), (2.25) reduces to [14,Corollary 12].
By imposing an additional condition, we get the following result which shows the extension of Theorem 2.3. We will leave the proof for reader.
Theorem 2.4. Along with the assumptions of Theorem 2.3, if h is bounded above by 1√2, then the following inequality holds:
2f∘g−1(u+mv2)≤1h(12α)h(2α−12α)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα2α)h(2α−tα2α)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.28) |
Remark 2.4. (i) For g(x)=x, (2.28) reduces to [14,Theorem 4].
(ii) For g(x)=xp, (2.28) reduces to [25,Theorem 4].
(iii) For α=1 along with the condition of (i), (2.28) reduces to [14,Corollary 9].
(iv) For α=1 along with the condition of (ii), (2.28) reduces to [25,Corollary 7].
Corollary 2.6. Applying h(t)=t in (2.25), the following fractional integral inequality for refined (α,m)-convex function with respect to a strictly monotone function g is obtained.
22α(2α−1)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β(2α(β+2α)−(β+α))22α(β+α)(β+2α)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.29) |
Corollary 2.7. Applying α=1 in (2.25), the following fractional integral inequality for refined (h−m)-convex function with respect to a strictly monotone function g is obtained.
1h2(12)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(t2)h(2−t2)tβ−1dt. | (2.30) |
Corollary 2.8. Applying α=1 and h(t)=ts, (2.25) gives the fractional integral inequality for refined (s,m)-convex function with respect to a strictly monotone function g in the following form:
22sf∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤2β−1β[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]B(s+β,1+s). | (2.31) |
Remark 2.5. If m=1 in (2.31), then the result for s-tgs convex function is obtained.
Corollary 2.9. Applying α=1 and h(t)=t, (2.25) gives the inequality for refined m-convex function with respect to a strictly monotone function g in the following form:
4f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β(β+3)4(β+1)(β+2)[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.32) |
Corollary 2.10. Applying α=1, (2.28) gives the following inequality:
2f∘g−1(u+mv2)≤1h2(12)f∘g−1(u+mv2)≤2βΓ(β+1)(mv−u)β[Iβ(u+mv2)+f∘g−1(mv)+mβ+1Iβ(u+mv2m)−f∘g−1(um)]≤β[f∘g−1(u)+2f∘g−1(v)+f∘g−1(um2)]∫10h(t2)h(2−t2)tβ−1dt≤12[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]. | (2.33) |
This section presents extensions of some results of Section 2 into k-fractional versions.
Theorem 3.1. Under the assumption of Theorem 2.1, the following inequality for k-fractional integral holds:
f∘g−1(u+mv2)h(12α)h(2α−12α)≤Γk(β+k)(mv−u)βk[kIβu+f∘g−1(mv)+mβk+1kIβv−f∘g−1(um)]≤βk[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα)h(1−tα)tβk−1dt, | (3.1) |
with β,k>0.
Proof. Using (1.7) and (1.8) after replacing β by βk in the proof of Theorem 2.1, we get the above inequality (3.1).
Theorem 3.2. Under the assumption of Theorem 2.3, for k>0, the following inequality holds for k-fractional integrals:
1h(12α)h(2α−12α)f∘g−1(u+mv2)≤2βkΓk(β+k)(mv−u)βk[kIβ(u+mv2)+f∘g−1(mv)+mβk+1Iβ(u+mv2m)−f∘g−1(um)]≤βk[f∘g−1(u)+2mf∘g−1(v)+m2f∘g−1(um2)]∫10h(tα2α)h(2α−tα2α)tβk−1dt, | (3.2) |
with β>0.
Proof. Using (1.7) and (1.8) after replacing β by βk in the proof of Theorem 2.3, we get the above inequality (3.2).
Remark 3.1. (i) For g(x)=x all the results of this section coincide with the results of [14,Section 3].
(ii) For g(x)=xp all the results of this section coincide with the results of [25,Section 3].
Next, we consider some examples of strictly monotone functions and provide the corresponding inequalities.
Example 3.1. For g(x)=xp, (2.13) gives the following results for refined (α,h−m)−p-convex function.
Case 3.1. For p>0, we have
f((up+mvp2)1p)h(12α)h(2α−12α)≤Γ(β+1)(mvp−up)β[Iβup+f(mvp)+mβ+1Iβ(mvp)−f(upm)]≤β[f(u)+2mf(v)+m2f(um2)]∫10h(tα)h(1−tα)tβ−1dt. | (3.3) |
Case 3.2. For p<0, we have
f((up+mvp2)1p)h(12α)h(2α−12α)≤Γ(β+1)(up−mvp)β[Iβup+f(mvp)+mβ+1Iβ(mvp)−f(upm)]≤β[f(u)+2mf(v)+m2f(um2)]∫10h(tα)h(1−tα)tβ−1dt. | (3.4) |
Example 3.2. For g(x)=ln(x) with g−1(x)=ex, (2.13) gives the following result:
f(eu+mv2)h(12α)h(2α−12α)≤Γ(β+1)(mv−u)β[Iβu+f(emv+mβ+1Iβv−f(eum)]≤β[f(eu)+2mf(ev)+m2f(eum2)]∫10h(tα)h(1−tα)tβ−1dt. | (3.5) |
Remark 3.2. (i) For h(t)=t, (3.3) reduces to [25,Corollary 2.1].
(ii) For α=1, (3.3) reduces to [25,Corollary 2.2].
(iii) For α=1 and h(t)=ts, (3.3) reduces to [25,Corollary 4].
Example 3.3. For g(x)=xp, (2.25) gives the following results for refined (α,h−m)−p-convex function.
Case 3.3. For p>0, we have
![]() |
(3.6) |
Case 3.4. For p<0, we have
![]() |
(3.7) |
Example 3.4. For g(x)=ln(x) with g−1(x)=ex, (2.25) gives the following result:
![]() |
(3.8) |
Remark 3.3. (i) For h(t)=t, (3.6) reduces to [25,Corollary 5].
(ii) For α=1, (3.6) reduces to [25,Corollary 6].
(iii) For α=1 and h(t)=ts, (3.6) reduces to [25,Corollary 8].
We have introduced a new class of convex functions that provides refinements of many known classes of functions related with convexity. Inequalities for Riemann-Liouville fractional integral operators have been established for this class of functions. Moreover, connection with already published results is given in the form of remarks and some examples are considered. Furthermore, for 0<h(t)<1, the refinements of the results of [13] can be obtained.
This research received funding support from the National Science, Research and Innovation Fund (NSRF), Thailand.
The authors declare that they have no competing interests.
[1] | V. D. Worp, H. Bart, E. S. Sena, et al., Hypothermia in animal models of acute ischaemic stroke: a systematic review and meta-analysis, Brain, 12 (2007), 3063-3074. |
[2] | H. Chen, M. Chopp, Z. G. Zhang, et al., The Effect of Hypothermia on Transient Middle Cerebral Artery Occlusion in the Rat, J. Cereb. Blood Flow. Metab., 4 (1992), 621-628. |
[3] | Y. H. Hwang, J. S. Jeon, Y. W. Kim, et al., Impact of immediate post-reperfusion cooling on outcome in patients with acute stroke and substantial ischemic changes, J. NeuroInt. Surg., 1 (2017), 21-25. |
[4] | S. Schwab, D. Georgiadis, J. Berrouschot, et al., Feasibility and safety of moderate hypothermia after massive hemispheric infarction, Stroke, 32 (2001), 2033-2035. |
[5] |
H. B. van der Worp, M. R. Macleod, P. M. W. Bath, et al., EuroHYP-1 investigators, 2014. EuroHYP-1: European multicenter, randomized, phase III clinical trial of therapeutic hypothermia plus best medical treatment vs. best medical treatment alone for acute ischemic stroke, Int. J. Stroke, 9 (2014), 642-645. doi: 10.1111/ijs.12294
![]() |
[6] | T. C. Wu and J. C. Grotta, Hypothermia for acute ischaemic stroke, Lancet Neurol., 3 (2013), 275-284. |
[7] | C. Wu, W. Zhao, H. An, et al., Safety, feasibility, and potential efficacy of intraarterial selective cooling infusion for stroke patients treated with mechanical thrombectomy. J. Cereb. Blood Flow. Metab., 12 (2018), 2251-2260. |
[8] | S. S. Song and P. D. Lyden, Overview of Therapeutic Hypothermia, Curr. Treat Options Neurol., 6 (2012), 541-548. |
[9] | G. Cattaneo, M. Schumacher, J. Wolfertz, et al., Open access combined selective cerebral hypothermia and mechanical artery recanalization in acute ischemic stroke: In vitro study of cooling performance, Am. J. Neuroradiol. 11 (2015), 2114-2120. |
[10] |
G. Cattaneo, M. Schumacher, C. Maurer, et al., Endovascular Cooling Catheter for Selective Brain Hypothermia: An Animal Feasibility Study of Cooling Performance, Am. J. Neuroradiol., 5 (2016), 885-891. doi: 10.3174/ajnr.A4625
![]() |
[11] | A. P. Avolio, Multi-branched model of the human arterial system, Med. Biol. Eng. Comput., 6 (1980), 709-718. |
[12] | M. Schwarz, Modellbasierte Operationsplanung und Überwachung hypothermer Patienten, KIT Scientific Publishing, 2009. |
[13] | M. Schwarz, M. W. Krueger, H. J. Busch, et al., Model-Based Assessment of Tissue Perfusion and Temperature in Deep Hypothermic Patients, IEEE Transact. Biomed. Eng., 7 (2010), 1577-1686. |
[14] | F. Umansky, S. M. Juarez, M. Dujovny, et al., Microsurgical anatomy of the proximal segments of the middle cerebral artery, J. Neurosurg., 3 (1984), 458-467. |
[15] | L. M. Parkes, W. Rashid, D. T. Chard, et al. Normal cerebral perfusion measurements using arterial spin labeling: Reproducibility, stability, and age and gender effects, Magnet. Reson. Med., 4 (2004), 736-743. |
[16] |
R. Fahrig, H. Nikolov, A. J. Fox, et al., A threedimensional cardiovascular flow phantom, Med. Phys., 8 (1999), 1589-1599. doi: 10.1118/1.598672
![]() |
[17] | J. S. Allen, H. Damasio and T. J. Grabowski, Normal neuroanatomical variation in the human brain: an MRI-volumetric study, Am. J. Phys. Anthropol., 4 (2002), 341-358. |
[18] | Y. Ge, R. I. Grossman, J. S. Babb, et al., Age-related total gray matter and white matter changes in normal adult brain, Part II: Quantitative magnetization transfer ratio histogram analysis, Am. J. Neuroradiol., 8 (2002), 1334-1341. |
[19] | Y. Taki, B. Thyreau, S. Kinomura, et al., Correlations among brain gray matter volumes, age, gender, and hemisphere in healthy individuals, PLOS ONE, 7 (2011), e22734. |
[20] | F. Mut, S. Wright, G. A. Ascoli, et al., Morphometric, geographic, and territorial characterization of brain arterial trees, Int. J. Numer. Method Biomed. Eng., 7 (2014), 755-766. |
[21] | J. Waschke and F. P. Sobotta, Atlas der Anatomie des Menschen: Kopf, Hals und Neuroanatomie. Urban und Fischer Verlag, 2010. |
[22] | W. C. Wu, S. C. Lin, K. L. Wang, et al., Measurement of cerebral white matter perfusion using pseudocontinuous arterial spin labeling 3t magnetic resonance imaging - an experimental and theoretical investigation of feasibility, PLoS ONE, 2013. |
[23] | K. Zilles and B. N. Tillmann, Anatomie Springer Verlag, 2010. |
[24] | N. Tariq and R. Khatri, Leptomeningeal collaterals in acute ischemic stroke, J. Vasc. Interv. Neurol., 1 (2008), 91-95. |
[25] | D. S. Liebeskind, Collateral circulation, Stroke, 8 (2003), 2279-2284. |
[26] | H. M. Vander Eecken and R. D. Adams, The anatomy and functional significance of the meningeal arterial anastomoses of the human brain, J. Neuropathol. Exper. neurol., 12 (1953), 132-157. |
[27] | A. Frydrychowski, A. Szarmach, B. Czaplewski, et al., Subarachnoid space: New tricks by an old dog. PloS one, 7 (2012), 37529. |
[28] | A. Bashkatov, E. Genina, Y. P. Sinichkin, et al., Glucose and mannitol diffusion in human dura mater, Biophys. J., 85 (2003), 3310-3318. |
[29] | H. Li, J. Ruan, Z. Xie, et al., Investigation of the critical geometric characteristics of living human skulls utilising medical image analysis techniques, Int. J. Veh. Saf.,2 (2007), 345-367. |
[30] | H. Hori, G. Moretti, A. Rebora, et al., The thickness of human scalp: normal and bald, J. Invest. Dermatol., 6 (1972), 396-369. |
[31] | M. Geerligs, Skin layer mechanics, PhD thesis, Department of Biomedical Engineering, TU Eindhoven, Eindhoven 2010. |
[32] | H. H. Pennes, Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm, J. Appl. Physiol., 1 (1948), 5-34. |
[33] | B. Pliskov, K. Mitra and M. Kaya, Simulation of scalp cooling by external devices for prevention of chemotherapy-induced alopecia, J. Therm. Biol., 56 (2016), 199-205. |
[34] | P. Hasgall, F. Di Gennaro, C. Baumgartner, et al., IT'IS Database for thermal and electromagnetic parameters of biological tissues, vol. 4.0, May 2018. Available from: itis.swiss/database 35. A.-A. Konstas, M. A. Neimark, A. F. Laine, et al., A theoretical model of selective cooling using intracarotid cold saline infusion in the human brain, J. Appl. Physiol., 4 (2007), 1329-1340. |
[35] | 36. L. Mcilvoy, Comparison of brain temperature to core temperature: a review of the literature. J. Neurosci. Nurs., 1 (2004), 23-31. |
[36] | 37. B. Karaszewski, J. M. Wardlaw, I. Marshall, et al., Measurement of brain temperature with magnetic resonance spectroscopy in acute ischemic stroke., Ann. Neurol., 4 (2006), 438-446. |
[37] | 38. T. C. Jackson and P. M. Kochanek. A New Vision for Therapeutic Hypothermia in the Era of Targeted Temperature Management: A Speculative Synthesis. Ther. Hypothermia Tem. Manag., 1 (2019), 13-47. |
[38] | 39. J. N. Stankowski and R. Gupta, Therapeutic targets for neuroprotection in acute ischemic stroke: lost in translation?, Antioxid. Redox Signal., 10 (2011), 1841-1851. |
[39] | 40. J. Wolfertz, S. Meckel, A. Guber, et al., Mathematical, numerical and in-vitro investigation of cooling performance of an intra-carotid catheter for selective brain hypothermia, Curr. Direct. Biomed. Eng., 1 (2015), -394. |
[40] | 41. J. Caroff, R. M. King, J. E. Mitchell, et al., Focal cooling of brain parenchyma in a transient large vessel occlusion model: proof-of-concept, J. NeuroInt. Surg., (2019), 1-6. |
[41] | 42. Y. Lutz, A. Loewe, S. Meckel, et al., Combined Local Hypothermia and Recanalization Therapy for Acute Ischemic Stroke: Estimation of Brain and Systemic Temperature Using an Energetic Numerical Model, Thermal Biol., 84 (2019), 316-322. |
[42] | 43. H. Lippert and R. Papst, Arterial variations in man: classification and frequency, J.F. BergmannVerlag Mnchen, 1985. |
[43] | 44. K. Cilliers, Anatomy of the middle cerebral artery: Cortical branches, branching pattern and anomalies, Trukish Neurosurg., 5(2017), 671-681. |
[44] | 45. M. A. Stefani, F. L. Schneider, A. C. H. Marrone, et al., Anatomic variations of anterior cerebral artery cortical branches, Clin. Anat., 4 (2000), 231-236. |
[45] | 46. K. Cilliers and B. Page, Detailed description of the anterior cerebral artery anomalies observed in a cadaver population, Ann. Anatomy-Anat. Anz., 208 (2016),1-8. |
[46] | 47. A. A. Zeal and A. L. Rhoton Jr., Microsurgical anatomy of the posterior cerebral artery, J. Neurosurg., 4 (1978), 534-559. |
[47] | 48. M. Pham and M. Bendszus, Facing time in ischemic stroke: an alternative hypothesis for collateral failure, Clin. Neuroradiol., 2 (2016), 141-151. |
1. | Haiying Wang, Dali Hu, 2024, Trapezium type inequality for the n-times differentiable a-preinvex functions, 979-8-3315-4024-1, 78, 10.1109/IHMSC62065.2024.00025 | |
2. | Haiying Wang, Zufeng Fu, Xiaolong Fu, Dali Hu, 2024, Hermite-Hadamard type inequality of the n-times differentiable E-preinvex functions, 979-8-3315-4024-1, 74, 10.1109/IHMSC62065.2024.00024 |