There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-h-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard (H.H) inequality along with a Jensen-type inequality for harmonically h-Godunova-Levin interval-valued functions (GL-IVFS). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.
Citation: Waqar Afzal, Waqas Nazeer, Thongchai Botmart, Savin Treanţă. Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation[J]. AIMS Mathematics, 2023, 8(1): 1696-1712. doi: 10.3934/math.2023087
[1] | Waqar Afzal, Khurram Shabbir, Thongchai Botmart, Savin Treanţă . Some new estimates of well known inequalities for $ (h_1, h_2) $-Godunova-Levin functions by means of center-radius order relation. AIMS Mathematics, 2023, 8(2): 3101-3119. doi: 10.3934/math.2023160 |
[2] | Waqar Afzal, Sayed M. Eldin, Waqas Nazeer, Ahmed M. Galal . Some integral inequalities for harmonical $ cr $-$ h $-Godunova-Levin stochastic processes. AIMS Mathematics, 2023, 8(6): 13473-13491. doi: 10.3934/math.2023683 |
[3] | Waqar Afzal, Khurram Shabbir, Thongchai Botmart . Generalized version of Jensen and Hermite-Hadamard inequalities for interval-valued $ (h_1, h_2) $-Godunova-Levin functions. AIMS Mathematics, 2022, 7(10): 19372-19387. doi: 10.3934/math.20221064 |
[4] | Waqar Afzal, Khurram Shabbir, Savin Treanţă, Kamsing Nonlaopon . Jensen and Hermite-Hadamard type inclusions for harmonical $ h $-Godunova-Levin functions. AIMS Mathematics, 2023, 8(2): 3303-3321. doi: 10.3934/math.2023170 |
[5] | Waqar Afzal, Najla M. Aloraini, Mujahid Abbas, Jong-Suk Ro, Abdullah A. Zaagan . Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes. AIMS Mathematics, 2024, 9(2): 5122-5146. doi: 10.3934/math.2024249 |
[6] | Waqar Afzal, Thongchai Botmart . Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes. AIMS Mathematics, 2023, 8(3): 7277-7291. doi: 10.3934/math.2023366 |
[7] | Iqra Nayab, Shahid Mubeen, Rana Safdar Ali, Faisal Zahoor, Muath Awadalla, Abd Elmotaleb A. M. A. Elamin . Novel fractional inequalities measured by Prabhakar fuzzy fractional operators pertaining to fuzzy convexities and preinvexities. AIMS Mathematics, 2024, 9(7): 17696-17715. doi: 10.3934/math.2024860 |
[8] | Sabila Ali, Rana Safdar Ali, Miguel Vivas-Cortez, Shahid Mubeen, Gauhar Rahman, Kottakkaran Sooppy Nisar . Some fractional integral inequalities via $ h $-Godunova-Levin preinvex function. AIMS Mathematics, 2022, 7(8): 13832-13844. doi: 10.3934/math.2022763 |
[9] | Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Bahaaeldin Abdalla, Ali Althobaiti . Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions. AIMS Mathematics, 2022, 7(1): 349-370. doi: 10.3934/math.2022024 |
[10] | Mujahid Abbas, Waqar Afzal, Thongchai Botmart, Ahmed M. Galal . Jensen, Ostrowski and Hermite-Hadamard type inequalities for $ h $-convex stochastic processes by means of center-radius order relation. AIMS Mathematics, 2023, 8(7): 16013-16030. doi: 10.3934/math.2023817 |
There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-h-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard (H.H) inequality along with a Jensen-type inequality for harmonically h-Godunova-Levin interval-valued functions (GL-IVFS). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.
The interval analysis discipline addresses uncertainty using interval variables in contrast to variables in the form of points, the calculation results are reported as intervals, preventing mistakes that may lead to false conclusions. Despite its long history, Moore [1], used interval analysis for the first time in 1969 to analyze automatic error reports. This led to an improvement in calculation performance, which attracted many scholars' attention. Due to their ability to be expressed as uncertain variables, intervals are commonly used in uncertain problems, such as computer graphics [2], decision-making analysis [3], multi-objective optimization [4], and error analysis [5]. Consequently, interval analysis has produced numerous excellent results, and interested readers can consult. [6,7,8].
Meanwhile, numerous disciplines, including economics, control theory, and optimization, use convex analysis and many scholars have studied it, see [9,10,11,12]. Recently, generalized convexity of interval-valued functions (IVFS) has received extensive research and has been utilized in a large number of fields and applications, see [13,14,15,16]. The (A, s)-convex and (A, s)-concave mappings describe the continuity of IVFS, as described by Breckner in [17]. Numerous inequalities have recently been established for IVFS. By applying the generalized Hukuhara derivative to IVFS, Chalco-Cano et al. [18] derived some Ostrowski-type inclusions. Costa [19], established Opial type inequalities for the generalized Hukuhara differentiable IVFS. In general, we can define a classical Hermite Hadamard inequality as follows:
η(t+u2)≤1u−t∫utη(ν)dν≤η(t)+η(u)2. | (1.1) |
Considering this inequality was the first geometrical interpretation of convex mappings in elementary mathematics, it has gained a lot of attention. The following are some variations and generalizations of this inequality, see [20,21,22,23]. Initially in 2007, Varoşanec [24] developed the notion of h-convex. Several authors have contributed to the development of inequalities based on H.H using h-convex functions, see [25,26,27,28]. The harmonically h-convex functions introduced by Noor [29], are important generalizations of convex functions. Here are some recent results relating to harmonically h-convexity, see [30,31,32,33,34,35]. At present, these results are derived from inclusion relations and interval LU-order relationships, both of which have significant flaws because these are partial order relations. It can be demonstrated the validity of the claim by comparing examples from the literature with those derived from these old relations. In light of this, determining how to use a total order relation to investigate convexity and inequality is crucial. As an additional observation, the interval differences between endpoints are much closer in examples than in these old partial order relations. Because of this, the ability to analyze convexity and inequalities using a total order relation is essential. Therefore, we will focus our entire paper on Bhunia et al. [36], (CR)- order relation. Using cr-order, Rahman [37], studied nonlinear constrained optimization problems with cr-convex functions. Based on the notions of cr-order relation, Wei Liu and his co-authors developed a modified version of H.H and Jensen-type inequalities for h-convex and harmonic h-convex functions by using center radius order relation, see [38,39].
Theorem 1.1 (See [38]). Let η:[t,u]→RI+. Consider h:(0,1)→R+ and h(12)≠0. If η∈SHX(cr-h,[t,u],RI+) and η∈ IR[t,u], then
12h(12)η(2tut+u)⪯crutu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]∫10h(x)dx. | (1.2) |
In addition, a Jensen-type inequality was also proved with harmonic cr-h-convexity.
Theorem 1.2 (See [38]). Let di∈R+, zi∈[t,u], η:[t,u]→RI+. If h is super multiplicative and non-negative function and η∈SHX(cr-h,[t,u],RI+). Then the inequality become as:
η(11Dk∑ki=1dizi)⪯crk∑i=1h(diDk)η(zi). | (1.3) |
Using the h-GL function, Ohud Almutairi and Adem Kiliman have proven the following result in 2019, see [40].
Theorem 1.3. Let η:[t,u]→R. If η is h-Godunova-Levin function and h(12)≠0. Then
h(12)2η(t+u2)≤1u−t∫utη(ν)dν≤[η(t)+η(u)]∫10dxh(x). | (1.4) |
This study is unique in that it introduces a notion of interval-valued harmonical h-Godunova-Levin functions that are related to a total order relation, called Center-Radius order, which is novel in the literature. By incorporating cr-interval-valued functions into inequalities, this article opens up a new avenue of research in inequalities. In contrast to classical interval-valued analysis, cr-order interval-valued analysis follows a different methodology. Based on the concept of center and radius, we calculate intervals as follows: tc=t_+¯t2 and tr=¯t−t_2, respectively, where ¯t and t_ are endpoints of interval t.
Inspired by. [15,34,38,39,41], This study introduces a novel class of harmonically cr-h-GL functions based on cr-order. First, we derived some H.H inequalities, then we developed the Jensen inequality using this new class. In addition, the study presents useful examples in support of its conclusions.
Lastly the paper is designed as follows: In section 2, preliminary information is provided. The key problems are described in section 3. There is a conclusion at the end of section 6.
Some notions are used in this paper that aren't defined in this paper, see [38,41]. The collection of intervals is denoted by RI of R, while the collection of all positive intervals can be denoted by R+I. For ν∈R, the scalar multiplication and addition are defined as
t+u=[t_,¯t]+[u_,¯u]=[t_+u_,¯t+¯u] |
νt=ν.[t_,¯t]={[νt_,μ¯t],ifν>0,{0},ifν=0,[ν¯t,νt_],ifν<0, |
respectively. Let t=[t_,¯t]∈RI, tc=t_+¯t2 is called center of interval t and tr=¯t−t_2 is said to be radius of interval t. In the case of interval t, this is the (CR) form
t=(t_+¯t2,¯t−t_2)=(tc,tr). |
An order relation between radius and center can be defined as follows.
Definition 2.1. (See [25]). Consider t=[t_,¯t]=(tc,tr), u=[u_,¯u]=(uc,ur)∈RI, then centre-radius order (In short cr-order) relation is defined as
t⪯cru⇔{tc<uc,tc≠uc,tc≤uc,tc=uc. |
Further, we represented the concept of Riemann integrable (in short IR) in the context of IVFS [39].
Theorem 2.1 (See [39]). Let φ:[t,u]→RI be IVF given by η(ν)=[η_(ν),¯η(ν)] for each ν∈[t,u] and η_,¯η are IR over interval [t,u]. In that case, we would call η is IR over interval [t,u], and
∫utη(ν)dν=[∫utη_(ν)dν,∫ut¯η(ν)dν]. |
All Riemann integrables (IR) IVFS over the interval should be assigned IR[t,u].
Theorem 2.2 (See [39]). Let η,ζ:[t,u]→R+I given by η=[η_,¯η], and ζ=[ζ_,¯ζ]. If η,ζ∈IR[t,u], and η(ν)⪯crζ(ν) ∀ν∈[t,u], then
∫utη(ν)dν⪯cr∫utζνdν. |
See interval analysis notations for a more detailed explanation, see [38,39].
Definition 2.2 (See [39]). Consider h:[0,1]→R+. We say that η:[t,u]→R+ is known harmonically h-convex function, or that η∈SHX(h,[t,u],R+), if ∀ t1,u1∈[t,u] and ν∈[0,1], we have
η(t1u1νt1+(1−ν)u1)≤ h(ν)η(t1)+h(1−ν)η(u1). | (2.1) |
If in (2.1) ≤ replaced with ≥ it is called harmonically h-concave function or η∈SHV(h,[t,u],R+).
Definition 2.3. (See [27]). Consider h:(0,1)→R+. We say that η:[t,u]→R+ is known as harmonically h-GL function, or that η∈SGHX(h,[t,u],R+), if ∀ t1,u1∈[t,u] and ν∈(0,1), we have
η(t1u1νt1+(1−ν)u1)≤η(t1)h(ν)+η(u1)h(1−ν). | (2.2) |
If in (2.2) ≤ replaced with ≥ it is called harmonically h-Godunova-Levin concave function or η∈SGHV(h,[t,u],R+).
Now let's look at the IVF concept with respect to cr-h-convexity.
Definition 2.4 (See [39]) Consider h:[0,1]→R+. We say that η=[η_,¯η]:[t,u]→R+I is called harmonically cr-h-convex function, or that η∈SHX(cr-h,[t,u],R+I), if ∀ t1,u1∈[t,u] and ν∈[0,1], we have
η(t1u1νt1+(1−ν)u1)⪯cr h(ν)η(t1)+h(1−ν)η(u1). | (2.3) |
If in (2.3) ⪯cr replaced with ⪰cr it is called harmonically cr-h-concave function or η∈SHV(cr-h,[t,u],R+I).
Definition 2.5. (See [39]) Consider h:(0,1)→R+. We say that η=[η_,¯η]:[t,u]→R+I is called harmonically cr-h-Godunova-Levin convex function, or that η∈SGHX(cr-h,[t,u],R+I), if ∀ t1,u1∈[t,u] and ν∈(0,1), we have
η(t1u1νt1+(1−ν)u1)⪯crη(t1)h(ν)+η(u1)h(1−ν). | (2.4) |
If in (2.4) ⪯cr replaced with ⪰cr it is called harmonically cr-h-Godunova-Levin concave function or η∈SGHV(cr-h,[t,u],R+I).
Remark 2.1.
(i) If h(ν)=1, in this case, Definition 2.5 becomes a harmonically cr-P-function [28].
(ii) If h(ν)=1h(ν), in this case, Definition 2.5 becomes a harmonically cr h-convex function [28].
(iii) If h(ν)=ν, in this case, Definition 2.5 becomes a harmonically cr-Godunova-Levin function [28].
(iv) If h(ν)=1νs, in this case, Definition 2.5 becomes a harmonically cr-s-convex function [28].
(v) If h(ν)=νs, in this case, Definition 2.5 becomes a harmonically cr-s-GL function [28].
Proposition 3.1. Define h1,h2:(0,1)→R+ functions that are non-negative and
1h2≤1h1,ν∈(0,1). |
If η∈SGHX(cr-h2,[t,u],RI+), then η∈SGHX(cr-h1,[t,u],RI+).
Proof. Since η∈SGHX(cr-h2,[t,u],RI+), then for all t1,u1∈[t,u],ν∈(0,1), we have
η(t1u1νt1+(1−ν)u1)⪯crη(t1)h2(ν)+η(u1)h2(1−ν) |
⪯crη(t1)h1(ν)+η(u1)h1(1−ν). |
Hence, η∈SGHX(cr-h1,[t,u],RI+).
Proposition 3.2. Let η:[t,u]→RI given by [η_,¯η]=(ηc,ηr). If ηc and ηr are harmonically h-GL over [t,u], then η is known as harmonically cr-h-GL function over [t,u].
Proof. Since ηc and ηr are harmonically h-GL over [t,u], then for each ν∈(0,1) and for all t1,u1∈[t,u], we have
ηc(t1u1νt1+(1−ν)u1)⪯crηc(t1)h(ν)+ηc(u1)h(1−ν), |
and
ηr(t1u1νt1+(1−ν)u1)⪯crηr(t1)h(ν)+ηr(u1)h(1−ν). |
Now, if
ηc(t1u1νt1+(1−ν)u1)≠ηc(t1)h(ν)+ηc(u1)h(1−ν), |
then for each ν∈(0,1) and for all t1,u1∈[t,u],
ηc(t1u1νt1+(1−ν)u1)<ηc(t1)h(ν)+ηc(u1)h(1−ν). |
Accordingly,
ηc(t1u1νt1+(1−ν)u1)⪯crηc(t1)h(ν)+ηc(u1)h(1−ν). |
Otherwise, for each ν∈(0,1) and for all t1,u1∈[t,u],
ηr(t1u1νt1+(1−ν)u1)≤ηr(t1)h(ν)+ηr(u1)h(1−ν)⇒η(t1u1νt1+(1−ν)u1)⪯crη(t1)h(ν)+η(u1)h(1−ν). |
Based on all the above, and Definition 2.1, this can be expressed as follows:
η(t1u1νt1+(1−ν)u1)⪯crη(t1)h(ν)+η(u1)h(1−ν) |
for each ν∈(0,1) and for all t1,u1∈[t,u].
This completes the proof.
This section developed the H.H inequalities for harmonically cr-h-GL functions.
Theorem 4.1. Consider h:(0,1)→R+ and h(12)≠0. Let η:[t,u]→RI+, if η∈SGHX(cr-h,[t,u],RI+) and η∈ IR[t,u], we have
[h(12)]2f(2tut+u)⪯crtuu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]∫10dxh(x). | (4.1) |
Proof. Since η∈SGHX(cr-h,[t,u],RI+), we have
h(12)η(2tut+u)⪯crη(tuxt+(1−x)u)+η((tu1−x)t+xu). |
On integration over (0,1), we have
h(12)η(2tut+u)⪯cr[∫10η(tuxt+(1−x)u)dx+∫10η(tu(1−x)t+xu)dx]=[∫10η_(tuxt+(1−x)u)dx+∫10η_(tu(1−x)t+xu)dx,∫10¯η(tuxt+(1−x)u)dx+∫10¯η(tu(1−x)t+xu)dx]=[2tuu−t∫utη_(ν)ν2dν,2tuu−t∫ut¯η(ν)ν2dν]=2tuu−t∫utη(ν)ν2dν. | (4.2) |
By Definition 2.5, we have
η(tuxt+(1−x)u)⪯crη(t)h(x)+η(u)h(1−x). |
On integration over (0, 1), we have
∫10η(tuxt+(1−x)u)dx⪯crη(t)∫10dxh(x)+η(u)∫10dxh(1−x). |
Accordingly,
utu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]∫10dxh(x). | (4.3) |
Adding (4.2) and (4.3), results are obtained as expected
h(12)2η(2tut+u)⪯crutu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]∫10dxh(x). |
Remark 4.1.
(i) If h(x)=1, in this case, Theorem 4.1 becomes result for harmonically cr- P-function:
12η(2tut+u)⪯crutu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]. |
(ii) If h(x)=1x, in this case, Theorem 4.1 becomes result for harmonically cr-convex function:
η(2tut+u)⪯crutu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]2. |
(iii) If h(x)=1(x)s, in this case, Theorem 4.1 becomes result for harmonically cr-s-convex function:
2s−1η(2tut+u)⪯crutu−t∫utη(ν)ν2dν⪯cr[η(t)+η(u)]s+1. |
Example 4.1. Let [t,u]=[1,2], h(x)=1x, ∀ x∈ (0,1). η:[t,u]→RI+ is defined as
η(ν)=[−1ν4+2,1ν4+3], |
where
h(12)2η(2tut+u)=η(43)=[431256,849256], |
utu−t∫utη(ν)ν2dν=2[∫21(2ν4−1ν6)dν,∫21(3ν4+1ν6)dν]=[258160,542160], |
[η(t)+η(u)]∫10dxh(x)=[478,1138]. |
As a result,
[431256,849256]⪯cr[258160,542160]⪯cr[478,1138]. |
Thus, proving the theorem above.
Theorem 4.2. Consider h:(0,1)→R+ and h(12)≠0. Let η:[t,u]→RI+, if η∈SGX(cr-h,[t,u],RI+) and η∈ IR[t,u], we have
[h(12)]24η(2tut+u)⪯cr△1⪯cr1u−t∫utη(ν)ν2dν⪯cr△2 |
⪯cr{[η(t)+η(u)][12+1h(12)]}∫10dxh(x), |
where
△1=[h(12)]4[η(4tu3t+u)+η(4tu3u+t)], |
△2=[η(2tut+u)+η(t)+η(u)2)]∫10dxh(x). |
Proof. Consider [t,t+u2], we have
η(4tut+3u)⪯crη(t2tut+uxt+(1−x)2tut+u )[h(12)]+η(t2tut+u(1−x)t+x2tut+u )[h(12)]. |
Integration over (0,1), we have
[h(12)]4η(4tuu+3t)⪯crutu−t∫2tut+uuη(ν)ν2dν. | (4.4) |
Similarly for interval [t+u2,u], we have
[h(12)]4η(4tut+3u)⪯crutu−t∫t2tut+uη(ν)ν2dν. | (4.5) |
Adding inequalities (4.4) and (4.5), we get
△1=[h(12)]4[η(4tuu+3t)+η(4utt+3u)]⪯crutu−t∫utη(ν)ν2dν. |
Now
[h(12)]24η(2tut+u)=[h(12)]24η(12(4tu3u+t)+12(4tu3t+u))⪯cr[h(12)]24[η(4tuu+3t)h(12)+η(4tu3u+t)h(12)]=[h(12)]4[η(4tuu+3t)+η(4tu3u+t)]=△1⪯crutu−t∫tuη(ν)ν2dν⪯cr12[η(t)+η(u)+2η(2tut+u)]∫10dxh(x)=△2⪯cr[η(t)+η(u)2+η(t)h(12)+η(u)h(12)]∫10dxh(x)⪯cr[η(t)+η(u)2+1h(12)[η(t)+η(u)]]∫10dxh(x)⪯cr{[η(t)+η(u)][12+1h(12)]}∫10dxh(x). |
Example 4.2. Thanks to example 4.1, we have
[h(12)]24η(2tut+u)=η(43)=[431256,849256], |
△1=12[η(85)+η(87)]=[66794096,138014096], |
△2=[η(1)+η(2)2+η(43)]∫10dxh(x), |
△2=[1935512,4465512], |
{[η(t)+η(u)][12+1h(12)]}∫10dxh(x)=[478,1138]. |
Thus, we obtain
[431256,849256]⪯cr[66794096,138014096]⪯cr[258160,542160]⪯cr[1935512,4465512]⪯cr[478,1138]. |
This proves the above theorem.
Theorem 4.3. Let η,ζ:[t,u]→RI+,h1,h2:(0,1)→R+ such that h1,h2≠0. If η∈SGHX(cr -h1,[t,u],RI+), ζ∈SGHX(cr-h2,[t,u],RI+) and η,ζ∈ IR[v,w] then, we have
utu−t∫utη(ν)ζ(ν)ν2dν⪯crM(t,u)∫101h1(x)h2(x)dx+N(t,u)∫101h1(x)h2(1−x)dx, | (4.6) |
where
M(t,u)=η(t)ζ(t)+η(u)ζ(u),N(t,u)=η(t)ζ(u)+η(u)ζ(t). |
Proof. Conider η∈SGHX(cr-h1,[t,u],RI+), ζ∈SGHX(cr-h2,[t,u],RI+) then, we have
η(tutx+(1−x)u)⪯crη(t)h1(x)+η(u)h1(1−x), |
ζ(tutx+(1−x)u)⪯crζ(t)h2(x)+ζ(u)h2(1−x). |
Then,
η(tutx+(1−x)u)ζ(tutx+(1−x)u) |
⪯crη(t)ζ(t)h1(x)h2(x)+η(t)ζ(u)h1(x)h2(1−x)+η(u)ζ(t)h1(1−x)h2(x)+η(u)ζ(u)h1(1−x)h2(1−x). |
Integration over (0, 1), we have
∫10η(tutx+(1−x)u)ζ(tutx+(1−x)u)dx=[∫10η_(tutx+(1−x)u)ζ_(tutx+(1−x)u)dx,∫10¯η(tutx+(1−x)u)¯ζ(tutx+(1−x)u)dx]=[utu−t∫utη_(ν)ζ_(ν)ν2dν,utu−t∫ut¯η(ν)¯ζ(ν)ν2dν]=utu−t∫utη(ν)ζ(ν)ν2dν⪯cr∫10[η(t)ζ(t)+η(u)ζ(u)]h1(x)h2(x)dx+∫10[η(t)ζ(u)+η(u)ζ(t)]h1(x)h2(1−x)dx. |
It follows that
utu−t∫utη(ν)ζ(ν)ν2dν⪯crM(t,u)∫101h1(x)h2(x)dx+N(t,u)∫101h1(x)h2(1−x)dx. |
Theorem is proved.
Example 4.3. Let [t,u]=[1,2], h1(x)=h2(x)=1x ∀ x∈ (0,1). η,ζ:[t,u]→RI+ be defined as
η(ν)=[−1ν4+2,1ν4+3],ζ(ν)=[−1ν+1,1ν+2]. |
Then,
utu−t∫utη(ν)ζ(ν)ν2dν=[282640,5986640],M(t,u)∫101h1(x)h2(x)dx=M(1,2)∫10x2dx=[3196,62996],N(t,u)∫101h1(x)h2(1−x)dx=N(1,2)∫10(x−x2)dx=[112,30796]. |
It follows that
[282640,5986640]⪯cr[3196,62996]+[112,30796]=[1332,394]. |
This proves the above theorem.
Theorem 4.4. Let η,ζ:[t,u]→RI+,h1,h2:(0,1)→R+ such that h1,h2≠0. If η∈SGHX(cr-h1,[t,u],RI+), ζ∈SGHX(cr-h2,[t,u],RI+) and η,ζ∈ IR[v,w] then, we have
h1(12)h2(12)2η(2tut+u)ζ(2tut+u)⪯crutu−t∫utη(ν)ζ(ν)ν2dμ+M(t,u)∫101h1(x)h2(1−x)dx+N(t,u)∫101h1(x)h2(x)dx. |
Proof. Since η∈SGHX(cr-h1,[t,u],RI+), ζ∈SGHX(cr-h2,[t,u],RI+), we have
η(2tut+u)⪯crη(tutx+(1−x)u)h1(12)+η(tut(1−x)+xu)h1(12),ζ(2tut+u)⪯crζ(tutx+(1−x)u)h2(12)+ζ(tut(1−x)+xu)h2(12). |
Then,
η(2tut+u)ζ(2tut+u)⪯cr1h1(12)h2(12)[η(tutx+(1−x)u)ζ(tutx+(1−x)u)+η(tut(1−x)+xu)ζ(tut(1−x)+xu)]+1h1(12)h2(12)[η(tutx+(1−x)u)ζ(tut(1−x)+xu)+η(tut(1−x)+xu)ζ(tutx+(1−x)u)]⪯cr1h1(12)h2(12)[η(tutx+(1−x)u)ζ(tutx+(1−x)u)+η(tut(1−x)+xu)ζ(tut(1−x)+xu)]+1h1(12)h2(12)[(η(t)h1(x)+η(u)h1(1−x))(ζ(u)h2(1−x)+ζ(u)h2(x))+(η(t)h1(1−x)+η(u)h1(x))(ζ(t)h2(x)+ζ(u)h2(1−x))]⪯cr1h1(12)h2(12)[η(tutx+(1−x)u)ζ(tutx+(1−x)u)+η(tut(1−x)+ux)ζ(tut(1−x)+ux)]+1h1(12)h2(12)[(1h1(x)h2(1−x)+1h1(1−x)h2(x))M(t,u)+(1h1(x)h2(x)+1h1(1−x)h2(1−x))N(t,u)]. |
Integration over (0,1), we have
∫10η(2tut+u)ζ(2tut+u)dx=[∫10η_(2tut+u)ζ_(2tut+u)dx,∫10¯η(2tut+u)¯ζ(2tut+u)dx]=η(2tut+u)ζ(2tut+u)dx⪯cr2h1(12)h2(12)[utu−t∫utη(ν)ζ(ν)ν2dν]+2h(12)h(12)[M(t,u)∫101h1(x)h2(1−x)dx+N(t,u)∫101h1(x)h2(x)dx]. |
Multiply both sides by h1(12)h2(12)2 above equation, we get required result
h1(12)h2(12)2η(2tut+u)ζ(2tut+u)⪯crutu−t∫utη(ν)ζ(ν)ν2dμ+M(t,u)∫101h1(x)h2(1−x)dx+N(t,u)∫101h1(x)h2(x)dx. |
Example 4.4. Let [t,u]=[1,2], h1(x)=h2(x)=1x, ∀ x∈ (0,1). η,ζ:[t,u]→RI+ be defined as
η(ν)=[−1ν4+2,1ν4+3],ζ(ν)=[−1ν+1,1ν+2]. |
Then,
h1(12)h2(12)2η(2tut+u)ζ(2tut+u)=2η(43)ζ(43)=[431512,9339512],utu−t∫utη(ν)ζ(ν)ν2dν=[282640,5986640],M(t,u)∫101h1(x)h2(1−x)dx=M(1,2)∫10(x−x2)dx=[31192,629192],N(t,u)∫101h1(x)h2(x)dx=N(1,2)∫10x2dx=[16,30748]. |
It follows that
[431512,9339512]⪯cr[282640,5986640]+[31192,629192]+[16,30748]=[123160,76140]. |
This proves the above theorem.
Theorem 5.1. Let di∈R+, zi∈[t,u]. If h is non-negative and super multiplicative function or η∈SGHX(cr-h,[t,u],RI+). Then the inequality become as :
η(11Dk∑ki=1dizi)⪯crk∑i=1[η(zi)h(diDk)], | (5.1) |
where Dk=∑ki=1di.
Proof. If k=2, inequality (5.1) holds. Assume that inequality (5.1) also holds for k−1, then
η(11Dk∑ki=1dizi)=η(1dkDkzk+∑k−1i=1diDkzi)=η(1dkDkzk+Dk−1Dk∑k−1i=1diDk−1zi)⪯crη(zk)h(dkDk)+η(∑k−1i=1diDk−1zi)h(Dk−1Dk)⪯crη(zk)h(dkDk)+k−1∑i=1[η(zi)h(diDk−1)]1h(Dk−1Dk)⪯crη(zk)h(dkDk)+k−1∑i=1[η(zi)h(diDk)]⪯crk∑i=1[η(zi)h(diDk)]. |
Therefore, the result can be proved by mathematical induction.
Remark 5.1.
(i) If h(x)=1, in this case, Theorem 5.1 becomes result for harmonically cr- P-function:
η(11Dk∑ki=1dizi)⪯crk∑i=1η(zi). |
(ii) If h(x)=1x, in this case, Theorem 5.1 becomes result for harmonically cr-convex function:
η(11Dk∑ki=1dizi)⪯crk∑i=1diDkη(zi). |
(iii) If h(x)=1(x)s, in this case, Theorem 5.1 becomes result for harmonically cr-s-convex function:
η(11Dk∑ki=1dizi)⪯crk∑i=1(diDk)sη(zi). |
This study presents a harmonically cr-h-GL concept for IVFS. Using this new concept, we study Jensen and H.H inequalities for IVFS. This study generalizes results developed by Wei Liu [38,39] and Ohud Almutairi [34]. Several relevant examples are provided as further support for our basic conclusions. It might be interesting to determine equivalent inequalities for different types of convexity in the future. Under the influence of this concept, a new direction begins to emerge in convex optimization theory. Using the cr-order relation, we will study automatic error analysis with intervals and apply harmonically cr-h-GL functions to optimize problems. Using this concept, we aim to benefit and advance the research of other scientists in various scientific disciplines.
This research received funding support from the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation (Grant number B05F650018).
The authors declare that there is no conflicts of interest in publishing this paper.
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