# q-Hermite–Hadamard Inequalities for Generalized Exponentially s,m;η-Preinvex Functions

Authors: Hua Wang; Humaira Kalsoom; Hüseyin Budak; Muhammad Idrees
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we introduce a new extension of classical convexity which is called generalized exponentiallys,m;η-preinvex functions. Also, it is seen that the new definition of generalized exponentially s,m;η-preinvex functions describes different new classes as special cases. To prove our main results, we derive a new qmκ2-integral identity for the twice qmκ2-differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the qmκ2-integral by utilizing differentiable exponentially s,m;η-preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.


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## Body

## 1. Introduction and Preliminaries

In mathematics, quantum calculus is equivalent to usual infinitesimal calculus without the concept of limits or the investigation of calculus without limits (quantum is from the Latin word “quantus,” and literally, it means how much, in Swedish “Kvant”). It has two major branches:q-calculus and h-calculus. And both of them were worked out by Cheung and Kac [1] in the early twentieth century. In the same era, Jackson started to work on quantum calculus or q-calculus, but Euler and Jacobi had already figured out this type of calculus. A number of studies have recently been widely used in the field of q-analysis, beginning with Euler, due to the vast necessity for mathematics that models of quantum computing q-calculus exist in the framework between physics and mathematics. In 2013, Tariboon and Ntouyas introduced the Dκ1q-difference operator [2, 3]. This inspired other researchers, and as a consequence, numerous novel results concerning quantum analogues of classical mathematical results have already been launched in the literature. In various mathematical fields, it has many applications, such as theory of numbers, combinations, orthogonal polynomials, basic hypergeometric functions and other subjects, quantum mechanics, physics, and the principle of relativity. Many important aspects of quantum calculus are covered in the articles by Humaira et al. [4–7]. The quantum calculus is currently a subfield of the more general scientific field of time-scale calculus. New developments have recently been made in the research and methodology of dynamic derivatives on time scales. The research offers a consolidation and application of traditional differential and difference equations. Moreover, it is a unification of the discrete theory with the continuous theory, from the theoretical perspective. Recently, in 2020, Bermudo et al. introduced the notion of the Dκ2q-derivative and integral [8]. For more details, see [9–15] and references cited therein.The discussion and application of convex functions has become a very rich source of motivational material in pure and applied science. This vision not only promoted new and profound results in many branches of mathematical and engineering sciences but also provided a comprehensive framework for the study of many problems. Many scholars have studied various classes of convex sets and convex functions; see [16, 17]. The concept of convexity has been extended in several directions, since these generalized versions have significant applications in different fields of pure and applied sciences. One of the convincing examples on extensions of convexity is the introduction of invex function, which was introduced by Hanson [18] Weir and Mond [19] explored the idea of preinvex functions and actualized it to the foundation of adequate optimality conditions and duality in nonlinear programming.The Hermite–Hadamard inequality was introduced by Hermite and Hadamard; see [20]. It is one of the most recognized inequalities in the theory of convex functional analysis, which is stated as follows.LetϜ:Q⊆ℛ⟶ℛ be a convex mapping and κ1,κ2∈Q with κ1<κ2. Then,(1)Ϝκ1+κ22≤1κ2−κ1∫κ1κ2Ϝϰdϰ≤Ϝκ1+Ϝκ22.IfϜ is concave, both inequalities hold in the reverse direction.The important objective of this paper is to introduce an exponentially generalized definition ofs,m;η-preinvex functions. Furthermore, the new qmκ2-integral identity is determined. By using this new identity, we proved many new estimates of bounds for it, essentially based on the concept of quantum calculus.


## 2. Preliminaries

In this section, we derive a new definition of the generalized exponentiallys,m;η-preinvex function. Also, we present all necessary concepts related to quantum calculus.First of all, letQ⊂ℛn be a nonempty set, Ϝ:Q⟶ℛ be a continuous function, and η:Q×Q×0,1⟶ℛ∖0 and ϑ:Q×Q×0,1⟶ℛn be two continuous functions.Definition 1.
A setQ⊆ℛn is supposed to be η-invex concerning η⋅,⋅,⋅ and ϑ⋅,⋅,⋅ with some fixed m∈0,1 if(2)mκ2+kηκ1,κ2,mϑκ1,κ2,m∈Q,for all κ1,κ2∈Q and k∈0,1.
Ifηκ1,κ2,m=1, the above equation is called the convex set, and ϑκ1,κ2,m=κ1−mκ2 is an invex set; however, the reverse is not possible.Example 1.
ConsiderQ=−3,−2∪−1,2 and(3)ϑκ1,κ2,m=κ1−mκ2if2≥κ2≥−1,2≥κ1≥−1,κ1−mκ2if−3≤κ2≤−2,−3≤κ1≤−2,−1−mκ2if−3≤κ2≤−2,−1≤κ1≤2,−3−mκ2if−1≤κ2≤2,−3≤κ1≤−2.
As one can see,Q is also an invex set for ϑ, but not a convex set.Definition 2.
A functionϜ:Q⟶ℛ is said to be a generalized exponentially s,m;η-preinvex function if there exist η⋅,⋅,⋅ and ϑ⋅,⋅,⋅, χ≥1, and nonpositive α such that(4)Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,m≤ksϜκ1χακ1+1−ksϜmκ2χαmκ2,for all κ1,κ2∈Q and k∈0,1 and for some fixed s,m∈0,1.Remark 1.
In Definition2,(1)
If we chooseα=0 or χ=1, then the definition of the generalized exponentially s,m;η-preinvex function is converted into the definition of the generalized s,m;η-preinvex function(2)
If we chooseα=0 and ηκ1,κ2,m=1, then we get the definition of s,m-preinvexity(3)
If we chooseα=0, ηκ1,κ2,m=1, and ϑκ1,κ2,m=κ1−mκ2, then we get the definition of s,m-convexity(4)
If we choosem=1 and ϑκ1,κ2,m=1, we get the definition in [21](4)
If we chooseχ=e, then we have the definition of exponentially s,m;η-preinvex functions, stated as followsDefinition 3.
A functionϜ:Q⟶ℛ is called exponentially s,m;η-preinvex if there exist η⋅,⋅,⋅, ϑ⋅,⋅.,⋅, and nonpositive α such that(5)Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,m≤ksϜκ1eακ1+1−ksϜmκ2eαmκ2,for all κ1,κ2∈Q and k∈0,1 and for some fixed s,m∈0,1.
Many researchers proved several results about the importance and development in the theory of exponentially convex functions and their applications. For more details, see [22–25] and references cited therein.
Jackson derived theq-Jackson integral in [12] from 0 to κ2 for 0<q<1 as follows:(6)∫0κ2Ϝϰdqϰ=1−qκ2∑n=0∞qnϜκ2qn,provided the sum converges absolutely.
Theq-Jackson integral in a generic interval κ1,κ2 was given by Jackson in [12] and defined as follows:(7)∫κ1κ2Ϝϰdqϰ=∫0κ2Ϝϰdqϰ−∫0κ1Ϝϰdqϰ.Definition 4.
(see [3]). We suppose that Ϝ:κ1,κ2⟶ℛ is an arbitrary function. Then, the qκ1-derivative of Ϝ at ϰ∈κ1,κ2 is defined as follows:(8)Dqκ1Ϝϰ=Ϝϰ−Ϝqϰ+1−qκ11−qϰ−κ1,ϰ≠κ1.
SinceϜ is an arbitrary function from κ1,κ2 to ℛ, Dqκ1Ϝκ1=limϰ⟶κ1Dqκ1Ϝϰ. The function Ϝ is said to be q-differentiable on κ1,κ2 if Dqκ1Ϝt exists for all ϰ∈κ1,κ2. If κ1=0 in (3), then Dq0Ϝϰ=DqϜϰ, where DqϜϰ is a familiar q-derivative of Ϝ at ϰ∈κ1,κ2 defined by the following expression (see [1]):(9)DqϜϰ=Ϝϰ−Ϝqϰ1−qϰ,ϰ≠0.Definition 5.
(see [8]). We suppose that Ϝ:κ1,κ2⟶ℛ is an arbitrary function; then, the qκ2-derivative of Ϝ at ϰ∈κ1,κ2 is defined as follows:(10)Dqκ2Ϝϰ=Ϝqϰ+1−qκ2−Ϝϰ1−qκ2−ϰ,ϰ≠κ2.Definition 6.
(see [3]). We suppose that Ϝ:κ1,κ2⟶ℛ is an arbitrary function; then, the qκ1-definite integral on κ1,κ2 is defined as follows:(11)∫κ1κ2Ϝϰdqκ1ϰ=1−qκ2−κ1∑n=0∞qnϜqnκ2+1−qnκ1=κ2−κ1∫01Ϝ1−tκ1+tκ2dqt.
In [10], Alp et al. established the qκ1-Hermite–Hadamard inequalities for convexity, which are defined as follows.Theorem 1.
LetϜ:κ1,κ2⟶ℛ be a convex differentiable function on κ1,κ2 and 0<q<1. Then, q-Hermite–Hadamard inequalities are as follows:(12)Ϝqκ1+κ22q≤1κ2−κ1∫κ1κ2Ϝϰdqκ1ϰ≤qϜκ1+Ϝκ22q.On the contrary, the following new description and related Hermite–Hadamard-form inequalities were given by Bermudo et al.Definition 7.
(see [8]). Let Ϝ:κ1,κ2⟶ℛ be an arbitrary function. Then, the qκ2-definite integral on κ1,κ2 is defined as(13)∫κ1κ2Ϝϰdqκ2ϰ=1−qκ2−κ1∑n=0∞qnϜqnκ1+1−qnκ2=κ2−κ1∫01Ϝtκ1+1−tκ2dqt.Theorem 2.
(see [8]). Let Ϝ:κ1,κ2⟶ℛ be a convex function on κ1,κ2 and 0<q<1. Then, q-Hermite–Hadamard inequalities are as follows:(14)Ϝκ1+qκ22q≤1κ2−κ1∫κ1κ2Ϝϰdqκ2ϰ≤Ϝκ1+qϜκ22q.From Theorems1 and 2, one can achieve the following inequalities.Corollary 1.
(see [8]). For any convex function Ϝ:κ1,κ2⟶ℛ and 0<q<1, we have(15)Ϝqκ1+κ22q+Ϝκ1+qκ22q≤1κ2−κ1∫κ1κ2Ϝϰdqκ1ϰ+∫κ1κ2Ϝϰdqκ2ϰ≤Ϝκ1+Ϝκ2,and(16)Ϝκ1+κ22≤12κ2−κ1∫κ1κ2Ϝϰdqκ1ϰ+∫κ1κ2Ϝϰdqκ2ϰ≤Ϝκ1+Ϝκ22.Alp and Sarikaya, by using the area of trapezoids, introduced the following generalized quantum integral which we will call asκ1Tq-integral.Definition 8.
(see [11]). Let Ϝ:κ1,κ2⟶ℛ be an arbitrary function. For ϰ∈κ1,κ2,(17)∫κ1κ2ϜξdqTκ1=1−qκ2−κ12q2q∑n=0∞qnϜqnκ2+1−qnκ1−Ϝκ2,where 0<q<1.Theorem 3.
(q¯-Hermite–Hadamard; see [11]). Let Ϝ:κ1,κ2⟶ℛ be a convex continuous function on κ1,κ2 and 0<q<1. Then, we have(18)Ϝκ1+κ22≤1κ2−κ1∫κ1κ2ϜϰdqTκ1ϰ≤Ϝκ1+Ϝκ22.Definition 9.
(see [11]). For any real number n, the q analogue of n is defined as(19)nq=1−qn1−q.Definition 10.
(see [11]). Let k,p>0. Then, Bqk,p is defined by(20)Bqk,p=∫01xk−11−qxdqqp−1x.


## 3. A Newqmκ2-Integral Identity

In this section, we present a newqmκ2-integral identity.Lemma 1.
Form∈0,1 with 0<q<1, let there be an arbitrary function Ϝ:Q⟶ℛ such that D2qϜmκ2 is qmκ2-integrable on Q. Then, one has(21)Lqmκ2κ1,κ2,m,x=Ϝmκ2+ηκ1,κ2,mϑκ1,κ2,m+qϜmκ22q−1ηκ1,κ2,mϑκ1,κ2,m∫mκ2+ηκ1,κ2,mϑκ1,κ2,mmκ2Ϝxmκ2dqx=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk.Proof.
We suppose that(22)∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk=∫01k1−qkqϜmκ2+kηκ1,κ2,mϑκ1,κ2,mk2q1−q2η2κ1,κ2,mϑ2κ1,κ2,m−2qϜmκ2+qkηκ1,κ2,mϑκ1,κ2,m+Ϝmκ2+q2kηκ1,κ2,mϑκ1,κ2,mk2q1−q2η2κ1,κ2,mϑ2κ1,κ2,mdqk=q∑n=0∞Ϝmκ2+qnηκ1,κ2,mϑκ1,κ2,m−2q∑n=0∞Ϝmκ2+qn+1ηκ1,κ2,mϑκ1,κ2,mq1−qη2κ1,κ2,mϑ2κ1,κ2,m+∑n=0∞Ϝmκ2+qn+2ηκ1,κ2,mϑκ1,κ2,mq1−qη2κ1,κ2,mϑ2κ1,κ2,m−qq1−qηκ1,κ2,mϑκ1,κ2,m∑n=0∞qnϜmκ2+qnηκ1,κ2,mϑκ1,κ2,mq1−q2η3κ1,κ2,mϑ3κ1,κ2,m−2q1−qηκ1,κ2,mϑκ1,κ2,m∑n=0∞qn+1Ϝmκ2+qn+1ηκ1,κ2,mϑκ1,κ2,mq21−q2η3κ1,κ2,mϑ3κ1,κ2,m+1−qηκ1,κ2,mϑκ1,κ2,m∑n=0∞qn+2Ϝmκ2+qn+2ηκ1,κ2,mϑκ1,κ2,mq31−q2η3κ1,κ2,mϑ3κ1,κ2,m=qϜmκ2+ηκ1,κ2,mϑκ1,κ2,m−Ϝmκ2−Ϝmκ2+qηκ1,κ2,mϑκ1,κ2,mq1−qη2κ1,κ2,mϑ2κ1,κ2,m+Ϝmκ2q1−qη2κ1,κ2,mϑ2κ1,κ2,m−2qq2η3κ1,κ2,mϑ3κ1,κ2,m∫mκ2+ηκ1,κ2,mϑκ1,κ2,mmκ2Ϝxdqmκ2x−q2+q−1Ϝmκ2+ηκ1,κ2,mϑκ1,κ2,mq21−qη2κ1,κ2,mϑ2κ1,κ2,m+Ϝmκ2+qηκ1,κ2,mϑκ1,κ2,mq1−qη2κ1,κ2,mϑ2κ1,κ2,m=Ϝmκ2+ηκ1,κ2,mϑκ1,κ2,m+qΛmκ2q2η2κ1,κ2,mϑ2κ1,κ2,m−2qq2η3κ1,κ2,mϑ3κ1,κ2,m∫mκ2+ηκ1,κ2,mϑκ1,κ2,mmκ2Ϝxdqmκ2x.
Multiplying both sides of the above equality byq2η2κ1,κ2,mϑ2κ1,κ2,m/2q, we get the required result.


## 4. Hermite–Hadamard Inequalities for Generalized Exponentiallys,m;η-Preinvex Functions

Theorem 4.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u≥1, then for some fixed s,m∈0,1, we have(23)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q2−1/uΩ1Dq2mκ2Ϝκ1χακ1u+Ω2Dq2mκ2Ϝmκ2χαmκ2u1/u,where(24)Ω1=Bqs+2,u+1,(25)Ω2=21−sBq2,u+1−Bqs+2,u+1.Proof.
By utilizing conditions of Lemma1 and the famous power mean inequality, we obtain(26)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01kdqk1−1/u×∫01k1−qkuDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q2−1/u∫01k1−qkuksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q2−1/u∫01k1−qkquksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u=q2η2κ1,κ2,mϑ2κ1,κ2,m2q2−1/uΩ1Dq2mκ2Ϝκ1χακ1u+Ω2Dq2mκ2Ϝmκ2χαmκ2u1/u,where(27)Ω1=∫01ks+11−qkqudqk=Bqs+2,u+1,and(28)Ω1=∫01k21−s−ks1−qkqudqk=21−sBq2,u+1−Bqs+2,u+1≥0due to 21−s−ks≥0 for all k∈0,1 and s∈0,1.
We proved our result.Theorem 5.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u>1 with p−1+u−1=1, then for some fixed s,m∈0,1, we obtain(29)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/p2,p+12q×2qDq2mκ2Ϝκ1/χακ1u+21−ss+2q−2qmκ2Dq2Ϝmκ2/χαmκ2u2qs+2q1/u.Proof.
By utilizing conditions of Lemma1 and the famous Hölder inequality, we obtain(30)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkdqpk1/p×∫01kDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkdqqpk1/p×∫01kksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/p2,p+12q×2qDq2mκ2Ϝκ1/χακ1u+21−ss+2q−2qDq2mκ2Ϝmκ2/χαmκ2u2qs+2q1/u.
This completes the proof.Theorem 6.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u>1 with p−1+u−1=1, then for some fixed s,m∈0,1, we obtain(31)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,m2qBqu+s+1,u+1mκ2Dq2Ϝκ1χακ1u+21−sBqu+1,u+1−Bqu+s+1,u+1mκ2Dq2Ϝκ1χαmκ2u1/u.Proof.
By utilizing conditions of Lemma1 and the famous Hölder inequality, we obtain(32)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫011dqk1/p×∫01ku1−qkuDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q×∫01ku1−qkuksDq2mκ2Ϝκ1χακ1u+1−ksmκ2Dq2Ϝmκ2χαmκ2udqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2qBqu+s+1,u+1mκ2Dq2Ϝκ1χακ1u+21−sBqu+1,u+1−Bqu+s+1,u+1mκ2Dq2Ϝmκ2χαmκ2u1/u,where(33)21−sBqu+1,u+1−Bqu+s+1,u+1≥0.
We proved our result.Theorem 7.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u>1 with p−1+u−1=1, then for some fixed s,m∈0,1, we obtain(34)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/pp+1,p+12q×ϑ1Dq2mκ2Ϝκ1χακ1u+ϑ2Dq2mκ2Ϝmκ2χαmκ2u1/u,where(35)ϑ1=1−q1−qs+1=1s+1q,ϑ2=1−q∑n=0∞qn1−qns.Proof.
By utilizing conditions of Lemma1 and the famous Hölder inequality, we obtain(36)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01kp1−qkpdqk1/p×∫01Dq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/pp+1,p+12q×∫01ksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u.
Applying the definition of quantum integral, we get(37)ϑ1=∫01ks0dqk=1−q1−qs+1=1s+1q,ϑ2=∫011−ks0dqk=1−q∑n=0∞qn1−qns.
This completes the proof.Theorem 8.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u>1 with p−1+u−1=1, then for some fixed s,m∈0,1, we obtain(38)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,m2qp+1q1/p×ω1mκ2Dq2Ϝκ1χακ1u+ω2Dq2mκ2Ϝmκ2χαmκ2u1/u,where(39)ω1=1−q∑n=0∞qns+11−qn+1u,ω2=1−q∑n=0∞qn1−qns1−qn+1u.Proof.
By utilizing conditions of Lemma1 and Hölder’s inequality, we have(40)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01kpdqk1/p×∫011−qkuDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,m2qp+1q1/p×∫011−qkuksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u.
Applying the definition of quantum integral, we get(41)ω1=∫01ks1−qku0dqk=1−q∑n=0∞qns+11−qn+1u,ω2=∫011−ks1−qku0dqk=1−q∑n=0∞qn1−qns1−qn+1u.
This completes the proof.Theorem 9.
We assume that the conditions of Lemma1 with χ≥1 and α∈ℛ hold. If Dq2mκ2Ϝu is a generalized exponentially s,m;η-preinvex function and u>1 with p−1+u−1=1, then for some fixed s,m∈0,1, we obtain(42)Lqmκ2κ1,κ2,m,x≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/p1,p+12q×σ1mκ2Dq2Ϝκ1χακ1u+σ2mκ2Dq2Ϝmκ2χαmκ2u1/u,where(43)σ1=1s+u+1q,σ2=1−q∑n=0∞qn1+u1−qns.Proof.
By utilizing conditions of Lemma1 and the famous Hölder inequality, we obtain(44)Lqmκ2κ1,κ2,m,x=q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫01k1−qkDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mdqk≤q2η2κ1,κ2,mϑ2κ1,κ2,m2q∫011−qkpdqk1/p×∫01kuDq2mκ2Ϝmκ2+kηκ1,κ2,mϑκ1,κ2,mudqk1/u≤q2η2κ1,κ2,mϑ2κ1,κ2,mBq1/p1,p+12q×∫01kuksDq2mκ2Ϝκ1χακ1u+1−ksDq2mκ2Ϝmκ2χαmκ2udqk1/u.
Applying the definition of quantum integral, we get(45)σ1=∫01ks+u0dqk=1s+u+1q,σ2=∫01ku1−ks0dqk=1−q∑n=0∞qn1+u1−qns.
This completes the proof.


## 5. Conclusion

In this article, we established the new definition of generalized exponentiallys,m;η-preinvex functions and proved a new modified qmκ2-integral identity. Using this new identity, we have been able to obtain new estimates of the quantum bounds applying the concept of generalized exponentially s,m;η-preinvex functions. It is worth to mention here that if we take χ=e, then all of the main results reduce to the results for exponentially s,m;η-preinvex functions. For further research, we could expand the inequality-based analysis to other fields, including the inequality-based theory, quantum calculus, machine learning, robotics, weather forecasting, and optimizations.



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*Source: 5577340-2021-04-19.xml*