
In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new qκ1-integral and qκ2-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard qκ1-integral and qκ2-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.
Citation: Humaira Kalsoom, Muhammad Amer Latif, Muhammad Idrees, Muhammad Arif, Zabidin Salleh. Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions[J]. AIMS Mathematics, 2021, 6(12): 13291-13310. doi: 10.3934/math.2021769
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In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new qκ1-integral and qκ2-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard qκ1-integral and qκ2-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.
In mathematics, the 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"). The renowned mathematician Euler was the genius who introduced the analysis q-calculus in the eighteenth century by integrating the parameter q into Newton's work of infinite series. At the beginning of the twentieth century, Jackson [1] has started a study of q-calculus and described quantum-definite integrals. The topic of quantum calculus has very long origins in the past. But to keep up with times, it has undergone rapid growth over the past few decades. I believe this strongly because it is a bridge between mathematics and physics, which is useful when dealing with physics. To get more information, please check the application and results of Ernst [2], Gauchman [3], and Kac and Cheung [4] in the theory of quantum calculus and theory of inequalities in quantum calculus. In previous papers, the authors Ntouyas and Tariboon [5,6] investigated how quantum-derivatives and quantum-integrals are solved over the intervals of the form [κ1,κ2]⊂R and set several quantum analogs. Our investigation here is motivated essentially by the fact that basic (or q-)Hölder inequality, Hermite-Hadamard inequality and Ostrowski inequality, Cauchy-Bunyakovsky-Schwarz, Gruss, Gruss-Cebysev and other integral inequalities that use classical convexity. Also, Noor et al. [7], Sudsutad et al. [8], and Zhuang et al. [9], played an active role in the study, and some integral inequalities have been established which give quantum analog for the right part of Hermite-Hadamard inequality by using q-differentiable convex and quasi-convex functions. Many mathematicians have done studies in q-calculus analysis, the interested reader can check [10,11,12,13,14,15,16,17,18].
Srivastava [19] presented (or q-)calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. There is also a clear connection between the classical q-analysis, which we used here, and the so-called (p,q)-analysis. We emphasize that the results for the q-analogues, which we discussed in this article for 0<q<1, can be easily (and probably trivially) converted into the corresponding results for the (p,q)-analogues (with 0<q<p≤1) by making a few obvious parametric and argument changes, with the additional parameter p being superfluous. Inspired and motivated by some of the above applications in the field of q-calculus. However, new q-Hermite-Hadamard-type inequalities for quantum integrals on finite intervals has not been studied yet. This gap is the motivation and inspiration for this research.
The discussion and application of convex functions have become a prosperous source of motivational material in pure and applied science. This vision promoted new and profound results in many branches of mathematical and engineering sciences and provided a comprehensive framework for the study of many problems. This discovery produced new and profound results in many mathematical and engineering sciences branches and provided a systematic structure for analyzing many issues in many fields. Many scholars have studied the various classes of convex sets and convex functions. A mapping K:I⊆R→R is considered convex if the mapping K satisfies the following inequality:
K((1−τ)κ1+τκ2)≤(1−τ)K(κ1)+τK(κ2) |
for all κ1,κ2∈I and τ∈[0,1].
One of the most famous inequalities in the theory of Convex Functional Analysis, Hermite-Hadamard established by Hermite and Hadamard in [20]. It has a very fascinating geometric representation with many significant applications. The extraordinary inequality states that, if K:I⊆R→R is a convex mapping on the interval I of real numbers and κ1,κ2∈I with κ1<κ2. Then,
K(κ1+κ22)≤1κ2−κ1κ2∫κ1K(τ)dτ≤K(κ1)+K(κ2)2. | (1.1) |
For K to be concave, both inequalities hold in the inverted direction. Many mathematicians have paid considerable attention to the Hermite-Hadamard inequality due to its quality and integrity in mathematical inequality. For significant developments, modifications, and consequences regarding the Hermite-Hadamard uniqueness property and general convex function definitions, the interested reader would like to refer to [21,22,23,24,25,26,27,28] and references therein.
It is noted that quasi-convex functions are a generalization of the convex function class since there are quasi-convex functions that are not convex. Weir et al. [29] introduced the concept of preinvex functions, which were then used in non-linear programming to describe appropriate optimal conditions and duality. Polyak [30] considered and studied the idea of strongly convex functions, which makes an essential contribution to the adaptation of most machine learning models that require the resolution of some form of optimization problem and areas concerned. Zu et al. [31] researched convergence by using iterative techniques based on the strong convex functional theory to resolve variational inequalities and equilibrium issues. Nikodem et al. discovered the new and innovative implementation of the inner product area's characterization with strongly convex functions in [32].
Throughout this paper, we are using continuous bifunctions μ(.,.):Rn×Rn→Rn and Iμ=[κ1,κ1+μ(κ2,κ1)]. Weir and Mond [29], has been introduced the definition of invex sets and preinvex mapping.
Definition 1.1. If Iμ⊂Rn, then Iμ⊂Rn is said to be invex set
κ1+τμ(κ2,κ1)∈Iμ, |
for all κ1,κ2∈Iμ,τ∈[0,1].
Note that, the invex set Iμ is also called μ-connected set. If μ(κ2,κ1)=κ2−κ1, then the invex set Iμ is a convex set, but the reverse is not true.
Definition 1.2. Let a mapping K:Iμ⊂Rn→R is called preinvex, if
K(κ1+τμ(κ2,κ1))≤(1−τ)K(κ1)+τK(κ2), |
for all κ1,κ2∈Iμ,τ∈[0,1].
Here, we would like to point out that Humaira et al. [11] has introduced and studied generalized higher-order strong preinvex functions, which play a crucial role in studying the theory of optimization and related fields.
Definition 1.3. A function K:Iμ⊂Rn→R is considered generalized higher-order strong preinvex function of order θ>0 with modulus χ≥0, if
K(κ1+τμ(κ2,κ1))≤(1−τ)K(κ1)+τK(κ2)−χτ(1−τ)||μ(κ2,κ1)||θ, |
for all κ1,κ2∈Iμ and all τ∈[0,1].
Properties that belong to generalized higher-order strongly preinvex functions are more robust versions of well-known properties of preinvex functions. Let us note the definition of the following generalized higher-order strongly quasi-preinvex functions.
Definition 1.4. [11] A function K:Iμ⊂Rn→R is considered generalized higher-order strong preinvex function of order θ>0 with modulus χ≥0, if
K(κ1+τμ(κ2,κ1))≤max{K(κ1),K(κ2)}−χτ(1−τ)||μ(κ2,κ1)||θ, |
for all κ1,κ2∈I and all τ∈[0,1].
Remark 1. The notion of generalized higher-order strongly quasi-preinvexity strengthens the concept of quasi-preinvexity.
Several fundamental inequalities that are well known in classical analysis, like {Hölder} inequality, Ostrowski inequality, Cauchy-Schwarz inequality, Grüess-Chebyshev inequality, Grüess inequality. Using classical convexity, other fundamental inequalities have been proven and applied to q-calculus.
Our objective is to develop new Hermite-Hadamard type inequalities by using quantum calculus and to support this claim graphically.
In this section, we discuss some required definitions of quantum calculus and important left and right sides bonds of quantum Hermite-Hadamard integral type inequalities.
[n]q=1−qn1−q=1+q+q2+...+qn−1,q∈(0,1). |
Jackson derived the q-Jackson integral in [1] from 0 to κ2 for q∈(0,1) as follows:
κ2∫0K(κ)cdqκ=(1−q)κ2∞∑n=0qnK(κ2qn) | (2.1) |
provided the sum converge absolutely.
The q-Jackson integral in a generic interval [κ1,κ2] was given by in [1] and defined as follows:
κ2∫κ1K(κ)cdqκ=κ2∫0K(κ)cdqκ−κ1∫0K(κ)cdqκ. |
Definition 2.1. [5] We suppose that K:[κ1,κ2]→R is an arbitrary function. Then qκ1-derivative of K at κ∈[κ1,κ2] is defined as follows:
cκ1DqK(κ)=K(κ)−K(qκ+(1−q)κ1)(1−q)(κ−κ1),κ≠κ1. | (2.2) |
Since K is a arbitrary function from [κ1,κ2] to R, thencκ1DqK(κ1)=limκ→κ1cκ1DqK(κ). The function K is called qκ1- differentiable on [κ1,κ2], if cκ1DqK(τ) exists for all κ∈[κ1,κ2]. If κ1=0 in (2.2), then c0DqK(κ)=DqK(κ), where cDqK(κ) is familiar qκ1-derivative of K at κ∈[κ1,κ2] defined by the expression (see [4])
cDqK(κ)=K(κ)−K(qκ)(1−q)κ,κ≠0. |
The lemma below is play key part to calculate qκ1-derivatives.
Lemma 2.2. [5] Taking ξ∈R and q∈(0,1), we have
κ1Dq(x−κ1)ξ=(1−qξ1−q)(x−κ1)ξ−1. |
Definition 2.3. [5] We suppose that K:[κ1,κ2]→R is an arbitrary function, then the qκ1-definite integral on [κ1,κ2] is described as below:
κ2∫κ1K(κ)cκ1dqκ=(1−q)(κ2−κ1)∞∑n=0qnK(qnκ2+(1−qn)κ1)=(κ2−κ1)1∫0K((1−τ)κ1+τκ2)cdqτ. | (2.3) |
for x∈I. If χ∈(κ1,x), then the definite qκ1-integral on I is described as:
∫xκ1K(x)κ1dqx=∫xκ1K(x)κ1dqx−∫χκ1K(x)κ1dqx=(x−κ1)(1−q)∞∑n=0qnK(qnx+(1−qn)κ1)+(χ−κ1)(1−q)∞∑n=0qnK(qnχ+(1−qn)κ1). |
If κ1=0 in (refA3), then we obtain the classical definite qκ1-integral which is proved in (see [6])
∫x0K(x)dqx=(1−q)x∞∑n=0qnK(qnx),x∈[0,∞). |
The following properties are very important in quantum calculus.
Theorem 2.4. [5] We suppose that K:I→R be a arbitrary function. Then
1. κ1Dq∫xκ1K(τ)κ1dqτ=K(x)−K(κ1);
2. ∫xχ κ1DqK(τ)κ1dqτ=K(x)−K(χ), χ∈(κ1,x).
The following is useful results for evaluating such qκ1-integrals.
Lemma 2.5. [5] The following formula holds for ζ∈R∖{−1} with q∈(0,1), then
∫σκ1(τ−κ1)ζκ1dqτ=(1−q1−qζ+1)(σ−κ1)ζ+1. |
In [10], Alp et al. established the qκ1 -Hermite-Hadamard inequalities for convexity, which is defined as follows:
Theorem 2.6. We suppose that K:[κ1,κ2]→R is a convex differentiable function on [κ1,κ2] and q∈(0,1). Then qκ1-Hermite-Hadamard inequalities are as follows:
K(qκ1+κ2[2]q)≤1κ2−κ1κ2∫κ1K(κ)[t]lκ1dqκ≤qK(κ1)+K(κ2)[2]q. | (2.4) |
In [18], Bermudo et al. established the qκ2-derivative, qκ2-integration and qκ2-Hermite-Hadamard inequalities for convexity, which is defined as follows:
Definition 2.7. [18] We suppose that K:[κ1,κ2]→R is an arbitrary function, then qκ2-derivative of K at κ∈[κ1,κ2] is defined as follows:
cκ2DqK(κ)=K(qκ+(1−q)κ2)−K(κ)(1−q)(κ2−κ),κ≠κ2. | (2.5) |
Since K is a arbitrary function from [κ1,κ2] to R, thencκ2DqK(κ2)=limκ→κ2cκ2DqK(κ). The function K is called qκ2- differentiable on [κ1,κ2], if cκ2DqK(τ) exists for all κ∈[κ1,κ2]. If κ2=0 in (2.5), then c0DqK(κ)=DqK(κ), where cDqK(κ) is familiar qκ2-derivative of K at κ∈[κ1,κ2] defined by the expression (see [1])
cDqK(κ)=K(qκ)−K(κ)(1−q)κ,κ≠0. |
Definition 2.8. We suppose that K:[κ1,κ2]→R is an arbitrary function. Then, the qκ2-definite integral on [κ1,κ2] is defined as:
κ2∫κ1K(κ)cκ2dqκ=(1−q)(κ2−κ1)∞∑n=0qnK(qnκ1+(1−qn)κ2)=(κ2−κ1)1∫0K(tκ1+(1−τ)κ2)cdqτ. |
Theorem 2.9. [18] We suppose that K:I→R is a continuous function. Then
1. κ2Dq∫κ2xK(τ)κ2dqτ=K(κ2)−K(x);
2. ∫χx κ2DqK(τ)κ2dqτ=K(χ)−K(x), χ∈(x,κ2).
Theorem 2.10. [18] We suppose that K:[κ1,κ2]→R be a convex function on [κ1,κ2] and q∈(0,1). Then, qκ2-Hermite-Hadamard inequalities are as follows:
K(κ1+qκ2[2]q)≤1κ2−κ1κ2∫κ1K(κ)[t]lκ2dqκ≤K(κ1)+qK(κ2)[2]q. | (2.6) |
From Theorem 2.6 and Theorem 2.10, one can the following inequalities:
Corollary 1. [18] For any convex function K:[κ1,κ2]→R and q∈(0,1), we have
K(qκ1+κ2[2]q)+K(κ1+qκ2[2]q)≤1κ2−κ1{κ2∫κ1K(κ)[t]lκ1dqκ+κ2∫κ1K(κ)[t]lκ2dqκ}≤K(κ1)+K(κ2) | (2.7) |
and
K(κ1+κ22)≤12(κ2−κ1){κ2∫κ1K(κ)[t]lκ1dqκ+κ2∫κ1K(κ)[t]lκ2dqκ}≤K(κ1)+K(κ2)2. | (2.8) |
Theorem 2.11. [7] Suppose that K:[κ1,κ1+μ(κ2,κ1)]⊂R→R is a qκ1-differentiable function on (κ1,κ1+μ(κ2,κ1)) such that κ1DqK being continuous and qκ1-integrable on [κ1,κ1+μ(κ2,κ1)] and q∈(0,1). If |κ1DqK|σ is preinvex function for σ≥1, then
|1μ(κ2,κ1)∫κ1+μ(k2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)[2]q(q(2+q+q3)[2]3q)1−1σ×[q(1+4q+q2)[3]q[2]3q|κ1DqK(κ1)|σ+q(1+3q2+2q3)[3]q[2]3q|κ1DqK(κ2)|σ]1σ | (2.9) |
Theorem 2.12. [7] Suppose that K:[κ1,κ1+μ(κ2,κ1)]⊂R→R is a qκ1-differentiable function on (κ1,κ1+μ(κ2,κ1)) such that κ1DqK being continuous and qκ1-integrable on [κ1,κ1+μ(κ2,κ1)] and q∈(0,1). If |κ1DqK|σ is quasi-preinvex function for σ≥1, then
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤q2μ(κ2,κ1)(2+q+q3)[2]4q(max{|κ1DqK(κ1)|σ,|κ1DqK(κ2)|σ})1σ. | (2.10) |
We are now providing some new Hermite-Hadamard-type inequalities for functions whose absolute value of first qκ1-, qκ2-derivatives are generalized higher-order strongly preinvex functions. To prove our main results, we will initially suggest the following useful lemmas.
Lemma 3.1. Suppose that K:[κ1,κ1+μ(κ2,κ1)]⊂R→R is a qκ1-differentiable function on (κ1,κ1+μ(κ2,κ1)) such that κ1DqK being continuous and qκ1-integrable on [κ1,κ1+μ(κ2,κ1)] and q∈(0,1), then the following identity holds
1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q=qμ(κ2,κ1)2×∫10∫10(ϵ−τ)[κ1DqK(κ1+τμ(κ2,κ1))−κ1DqK(κ1+ϵμ(κ2,κ1))]dqτ0dqϵ. | (3.1) |
Proof. By using Definition 2.1 and Definition 2.3, we have
∫10∫10(ϵ−τ))[κ1DqK(κ1+τμ(κ2,κ1))−κ1DqK(κ1+ϵμ(κ2,κ1))]dqτdqϵ=∫10∫10(ϵ−τ)[K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))(1−q)μ(κ2,κ1)τ−K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))(1−q)μ(κ2,κ1)ϵ]dqτdqϵ=∫10∫10ϵ[K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))](1−q)μ(κ2,κ1)τdqτdqϵ−∫10∫10K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))(1−q)μ(κ2,κ1)dqτdqϵ−∫10∫10K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))(1−q)μ(κ2,κ1)dqτdqϵ+∫10∫10τ[K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))](1−q)μ(κ2,κ1)ϵdqτdqϵ. | (3.2) |
We observe that
∫10∫10ϵ[K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))](1−q)μ(κ2,κ1)τdqτ0dqϵ=∫10ϵdqϵ∫10K(κ1+τμ(κ2,κ1))(1−q)μ(κ2,κ1)τ0dqτ−∫10ϵdqϵ∫10K(κ1+qτμ(κ2,κ1))(1−q)μ(κ2,κ1)τ dqτ=(1−q)μ(κ2,κ1)∞∑n=0q2n[∞∑n=0K(κ1+qnμ(κ2,κ1))−∞∑n=0K(κ1+qn+1μ(κ2,κ1))]=1[2]qμ(κ2,κ1)[∞∑n=0K(κ1+qnμ(κ2,κ1))−∞∑n=1K(κ1+qnμ(κ2,κ1))]=K(κ1+μ(κ2,κ1))−K(κ1)[2]qμ(κ2,κ1), | (3.3) |
and
∫10∫10K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))(1−q)μ(κ2,κ1)dqτdqϵ=∫10dqτ∫10K(κ1+ϵμ(κ2,κ1))(1−q)μ(κ2,κ1)0dqϵ−∫10dqτ∫10K(κ1+qϵμ(κ2,κ1))(1−q)μ(κ2,κ1) dqϵ=(1−q)μ(κ2,κ1)∞∑n=0qn[∞∑n=0qnK(κ1+qnμ(κ2,κ1))−∞∑n=0qnK(κ1+qn+1μ(κ2,κ1))]=1μ(κ2,κ1)∞∑n=0qnK(κ1+qnμ(κ2,κ1))−1μ(κ2,κ1)∞∑n=0qnK(κ1+qn+1μ(κ2,κ1))=1μ(κ2,κ1)[∞∑n=0qnK(κ1+qnμ(κ2,κ1))−1q∞∑n=0qn+1K(κ1+qn+1μ(κ2,κ1))]=−1qμ(κ2,κ1)2∫κ1+μ(κ2,κ1)κ1K(x) κ1dqx+K(κ1+μ(κ2,κ1))qμ(κ2,κ1). | (3.4) |
Similarly
∫10∫10K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))(1−q)μ(κ2,κ1)dqτdqϵ=∫10dqϵ∫10K(κ1+τμ(κ2,κ1))−K(κ1+qτμ(κ2,κ1))(1−q)μ(κ2,κ1)dqτ=−1qμ(κ2,κ1)2∫κ1+μ(κ2,κ1)κ1K(x) κ1dqx+K(κ1+μ(κ2,κ1))qμ(κ2,κ1), | (3.5) |
and
∫10∫10τ[K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))](1−q)μ(κ2,κ1)ϵdqτ0dqϵ=∫10τdqτ∫10K(κ1+ϵμ(κ2,κ1))−K(κ1+qϵμ(κ2,κ1))(1−q)μ(κ2,κ1)ϵdqϵ=K(κ1+μ(κ2,κ1))−K(κ1)[2]qμ(κ2,κ1). | (3.6) |
The equalities (3.3)–(3.6), give
∫10∫10(ϵ−τ)[κ1DqK(κ1+τμ(κ2,κ1))−κ1DqK(κ1+ϵμ(κ2,κ1))] dqτdqϵ=2qμ(κ2,κ1)2∫κ1+μ(κ2,κ1)κ1K(x) κ1dqx−2K(κ1+μ(κ2,κ1))qμ(κ2,κ1)+2[K(κ1+μ(κ2,κ1))−K(κ1)][2]qμ(κ2,κ1). | (3.7) |
Multiplying both sides of (3.7) by qμ(κ2,κ1)2, we get (3.1).
Lemma 3.2. Suppose that K:[κ2+μ(κ1,κ2),κ2]⊂R→R is a qκ2-differentiable function on (κ2+μ(κ1,κ2),κ2) such that κ2DqK being continuous and qκ2-integrable on [κ2+μ(κ1,κ2),κ2] with q∈(0,1) and μ(κ2,κ1)=−μ(κ1,κ2)>0, then the following identity holds
1μ(κ1,κ2)∫κ2κ2+μ(κ1,κ2)K(x)κ2dqx−K(κ2+μ(κ1,κ2))+qK(κ2)[2]q=qμ(κ1,κ2)2×∫10∫10(ϵ−τ)[κ2DqK(κ2+τμ(κ1,κ2))−κ2DqK(κ2+ϵμ(κ1,κ2))]dqτ0dqϵ. | (3.8) |
Proof. The proof is directly followed by Definition 2.7 and Definition 2.8. We omit the details.
Theorem 3.3. If we assume all the conditions of Lemma 3.1, then the following inequality, shows that |κ1DqK|σ is generalized higher-order strongly preinvex functions of order θ>0 with modulus χ≥0 on [κ1,κ1+μ(κ2,κ1)] for σ≥1, then
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)[ρ3(q)]1−1σ×[ρ1(q)|κ1DqK(κ1)|σ+ρ2(q)|κ1DqK(κ2)|σ−χμ(κ2,κ1)θρ4(q)]1σ, | (3.9) |
where
ρ1(q)=q(2q2−q+1)q5+2q4+3q3+3q2+2q+1, |
ρ2(q)=qq4+q3+2q2+q+1, |
ρ3(q)=2qq3+2q2+2q+1, |
and
ρ4(q)=q2(q4+q3+q2−q+1)q9+3q8+6q7+9q6+11q5+11q4+9q3+6q2+3q+1. |
Proof. Taking modulus on Eq (3.1) and using the power-mean inequality, we have
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)2(∫10∫10|ϵ−τ|dqτdqϵ)1−1σ×{(∫10∫10|ϵ−τ||κ1DqK(κ1+τμ(κ2,κ1))|σdqτdqϵ)1σ+(∫10∫10|ϵ−τ||κ1DqK(κ1+ϵμ(κ2,κ1))|σ dqτdqϵ)1σ}. | (3.10) |
Since |κ1DqK|σ is generalized higher-order strongly preinvex function for σ≥1, we have
∫10∫10|ϵ−τ||κ1DqK(κ1+τμ(κ2,κ1))|σdqτ0dqϵ≤|κ1DqK(κ1)|σ∫10∫10|ϵ−τ|(1−τ)0dqτdqϵ+|κ1DqK(κ2)|σ×∫10∫10|ϵ−τ|τdqτdqϵ−χμ(κ2,κ1)θ∫10∫10|ϵ−τ|τ(1−τ)dqτdqϵ, | (3.11) |
by using Definition 2.1 and Definition 2.3
ρ1(q)=∫10|ϵ−τ|(1−τ)0dqτdqϵ=∫10∫10[−2q2ϵ3[2]q[3]q+2qϵ2[2]q−qϵ[2]qq2[2]q[3]q]dqϵ=q(2q2−q+1)q5+2q4+3q3+3q2+2q+1, | (3.12) |
ρ2(q)=∫10∫10|ϵ−τ|τ0dqτdqϵ=∫10[2q2ϵ3[2]q[3]q−ϵ[2]q+1[3]q]dqϵ=qq4+q3+2q2+q+1, | (3.13) |
and
ρ4(q)∫10∫10|ϵ−τ|τ(1−τ)dqτ0dqϵ=∫10(−2∫ϵ0(τ−ϵ)τ(1−τ)dqτ+∫10(τ−ϵ)τ(1−τ)dqτ)0dqϵ=q2(q4+q3+q2−q+1)q9+3q8+6q7+9q6+11q5+11q4+9q3+6q2+3q+1. | (3.14) |
Using (3.12)–(3.14) in (3.11) and we get the resulting inequality
∫10∫10|ϵ−τ||κ1DqK(κ1+τμ(κ2,κ1))|σdqτ0dqϵ≤|κ1DqK(κ1)|σρ1(q)+|κ1DqK(κ2)|σρ2(q)−χμ(κ2,κ1)θρ4(q). | (3.15) |
Similarly, we also observe that
∫10∫10|ϵ−τ||κ1DqK(κ1+ϵμ(κ2,κ1))|σdqτ0dqϵ≤|κ1DqK(κ1)|σ∫10∫10|ϵ−τ|(1−ϵ)0dqτdqϵ+|κ1DqK(κ2)|σ∫10∫10|ϵ−τ|ϵdqτdqϵ−χμ(κ2,κ1)θ∫10∫10|ϵ−τ|τ(1−τ)dqτ0dqϵ=ρ1(q)|κ1DqK(κ1)|σ+ρ2(q)|κ1DqK(κ2)|σ−χμ(κ2,κ1)θρ4(q). | (3.16) |
We also have
ρ3(q)=∫10∫10|ϵ−τ|dqτ0dqϵ=∫10(−2∫ϵ0(τ−ϵ)dqτ+∫10(τ−ϵ)dqτ)0dqϵ=∫10(2qϵ2[2]q−ϵ+1[2]q)dqϵ=2q[2]q[3]q, | (3.17) |
Applying (3.15)–(3.17) in (3.10), we obtain the desired inequality.
Corollary 2. If σ=1 together with the assumptions of Theorem 3.3, we obtain
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)[ρ1(q)|κ1DqK(κ1)|+ρ2(q)|κ1DqK(κ2)|−χμ(κ2,κ1)θρ4(q)], | (3.18) |
where ρ1(q),ρ2(q) and ρ4(q) are defined in Theorem 3.3.
Corollary 3. As q→1− in Theorem 3.3, we get the inequality
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)dx−K(κ1)+K(κ1+μ(κ2,κ1))2|≤μ(κ2,κ1)(13)1−1σ[|K′(κ1)|σ+|K′(κ2)|σ6−χμ(κ2,κ1)θ20]1σ. | (3.19) |
Corollary 4. Suppose that the assumptions of Theorem 3.3 with σ=1 and letting q→1−, we obtain the inequality
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)dx−K(κ1)+K(κ1+μ(κ2,κ1))2|≤μ(κ2,κ1)[|K′(κ1)|+|K′(κ2)|6−χμ(κ2,κ1)θ20]. | (3.20) |
Theorem 3.4. If we assume all the conditions of Lemma 3.1, then the following inequality, shows that |κ1DqK|σ is generalized higher-order strongly preinvex function of order θ>0 with modulus χ≥0 on [κ1,κ1+μ(κ2,κ1)] for 1p+1σ=1, then
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)[ρ3(p,q)]1−1p |
×(q|κ1DqK(κ1)|σ+|κ1DqK(κ2)|σ[2]q−χμ(κ2,κ1)θq2[2]q[3]q)1σ, | (3.21) |
where
ρ3(p,q)=(q−1)2(qp+1−1)∞∑m=0(−1)m−1(3+qp−m+1−qm+1−2qp+1−qp+2)p(p−1)⋯(p−m+1)m!(qp−m+1+1)(qm+1−1). |
Proof. Taking modulus on Eq (3.1) and using Hölder's inequality, we have
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)2(∫10∫10|ϵ−τ|pdqτdqϵ)1−1p q×{(∫10∫10|κ1DqK(κ1+τμ(κ2,κ1))|σdqτ0dqϵ)1σ+(∫10∫10|κ1DqK(κ1+ϵμ(κ2,κ1))|σdqτdqϵ)1σ}. | (3.22) |
We now evaluate the integrals involved in (3.22). We observe that
∫10∫10|ϵ−τ|pdqτdqϵ=∫10(∫ϵ0(ϵ−τ)p dqτ)dqϵ+∫10(∫1ϵ(τ−ϵ)p0dqτ)dqϵ=∫10(∫ϵ0(ϵ−τ)pdqτ)dqϵ+∫10(∫ϵ0(τ−ϵ)p0dqτ)dqϵ+∫10(∫10(τ−ϵ)pdqτ)dqϵ, | (3.23) |
∫10(∫ϵ0(ϵ−τ)p0dqτ)dqϵ=1−q1−qp+1[1−p1[2]q+p(p−1)2!1[3]q−⋯]=(1−q)21−qp+1∞∑m=0(−1)m−1p(p−1)⋯(p−m+1)m!(1−qm+1), | (3.24) |
∫10(∫ϵ0(τ−ϵ)p0dqτ)dqϵ=∫10∫1qτ(τ−ϵ)pdqϵdqτ=∫10∫10(τ−ϵ)pdqϵ0dqτ−∫10∫qτ0(τ−ϵ)pdqϵdqτ=(1−q)2∞∑m=0(−1)m−1p(p−1)⋯(p−m+1)m!([2]p−m+1q)(1−qm+1)−q(1−q)21−qp+1∞∑m=0(−1)m−1qmp(p−1)⋯(p−m+1)m!(1−qm+1), | (3.25) |
and
∫10(∫10(τ−ϵ)pdqτ)dqϵ=∫10(∫10(τ−ϵ)pdqϵ)dqτ=(1−q)2∞∑m=0(−1)m−1p(p−1)⋯(p−m+1)m!([2]p−m+1q)(1−qm+1). | (3.26) |
Using the generalized higher-order strongly preinvexity of |κ1DqK|σ on [kappa1,κ1+μ(κ2,κ1)], we obtain
∫10∫10|κ1DqK(κ1+τμ(κ2,κ1))|σdqτdqϵ≤|κ1DqK(κ1)|σ∫10(1−τ)dqτ+|κ1DqK(κ2)|σ∫10τdqτ−χμ(κ2,κ1)θ∫10∫10(1−τ)τdqτdqϵ=q|κ1DqK(κ1)|σ+|κ1DqK(κ2)|σ[2]q−χμ(κ2,κ1)θq2[2]q[3]q. | (3.27) |
and similarly, we get
∫10∫10|κ1DqK(κ1+ϵμ(κ2,κ1))|σdqτdqϵ=q|κ1DqK(κ1)|σ+|κ1DqK(κ2)|σ[2]q−χμ(κ2,κ1)θq2 [2]q[3]q. | (3.28) |
Making use of (3.23) and (3.28) in (3.22), we get the required result.
Theorem 3.5. If we assume all the conditions of lemma 3.1, then the following inequality, shows that |κ1DqK|σ is generalized higher-order strongly quasi-preinvex function of order θ>0 with modulus χ≥0 on [κ1,κ1+μ(κ2,κ1)] for σ≥1, then
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)(ρ3(q))1−1σ(ρ3(q)ρ5(q)−χμ(κ2,κ1)θρ4(q))1σ, | (3.29) |
where
ρ5(q)=max{|κ1DqK(κ1)|σ,|κ1DqK(κ2)|σ}, |
ρ3(q) and ρ4(q) are defined in Theorem 3.3.
Proof. Taking modulus on equation (3.1) and using the power-mean inequality, we have
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)2(∫10∫10|ϵ−τ|dqτdqϵ)1−1σ×{(∫10∫10|ϵ−τ||κ1DqK(κ1+τμ(κ2,κ1))|σdqτdqϵ)1σ+(∫10∫10|ϵ−τ||κ1DqK(κ1+ϵμ(κ2,κ1))|σdqτdqϵ)1σ}. | (3.30) |
By using the generalized higher-order strongly quasi-preinvexity of |κ1DqK|σ on σ≥1, we obtain
|κ1DqK(κ1+τμ(κ2,κ1))|σ≤max{|κ1DqK(κ1)|σ,|κ1DqK(κ2)|σ}−χμ(κ2,κ1)θτ(1−τ) | (3.31) |
and
|κ1DqK(κ1+ϵμ(κ2,κ1))|σ≤max{|κ1DqK(κ1)|σ,|κ1DqK(κ2)|σ}−χμ(κ2,κ1)θϵ(1−ϵ), | (3.32) |
for all 0≤τ,ϵ≤1.
Applying (3.14), (3.17), (3.31) and (3.32) in (3.30), we get the desired result.
Corollary 5. Letting σ=1 in Theorem 3.5, we obtain
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dqx−qK(κ1)+K(κ1+μ(κ2,κ1))[2]q|≤qμ(κ2,κ1)(ρ3(q)ρ6(q)−χμ(κ2,κ1)θρ4(q)), | (3.33) |
where ρ6(q)=max{|κ1DqK(κ1)|,|κ1DqK(κ2)|}.
Corollary 6. Letting q→1− in Theorem 3.5, we obtain
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dx−K(κ1)+K(κ1+μ(κ2,κ1))2|≤μ(κ2,κ1)(13)1−1σ(ρ7(1)3−χμ(κ2,κ1)θ20)1σ, | (3.34) |
where ρ7(1)=max{|κ1DK(κ1)|σ,|κ1DK(κ2)|σ}.
Corollary 7. Letting q→1− in Theorem 3.5 together with σ=1, we obtain
|1μ(κ2,κ1)∫κ1+μ(κ2,κ1)κ1K(x)κ1dx−K(κ1)+K(κ1+μ(κ2,κ1))2|≤μ(κ2,κ1)(ρ8(1)3−χμ(κ2,κ1)θ20), | (3.35) |
where ρ8(1)=max{|κ1DK(κ1)|,|κ1DK(κ2)|}.
Theorem 3.6. If we assume all the conditions of Lemma 3.2, then the following inequality, shows that |κ2DqK|σ is generalized higher-order strongly preinvex function of order θ>0 with modulus χ≥0 on [κ2+μ(κ1,κ2),κ2] for σ≥1, then
|1μ(κ1,κ2)∫κ2κ2+μ(κ1,κ2)K(x)κ2dqx−K(κ2+μ(κ1,κ2))+qK(κ2)[2]q|≤qμ(κ1,κ2)[ρ3(q)]1−1σ×[ρ2(q)|κ2DqK(κ1)|σ+ρ1(q)|κ2DqK(κ2)|σ−χμ(κ1,κ2)θρ4(q)]1σ, | (3.36) |
where ρ1(q), ρ2(q), ρ3(q) and ρ4(q) are defined in Theorem 3.3.
Proof. The desired inequality (3.36) can be obtained by applying the strategy used in the proof of Theorem 3.3 and taking into account the Lemma 3.8.
Corollary 8. If σ=1 together with the assumptions of Theorem 3.6, we obtain
|1μ(κ1,κ2)∫κ2κ2+μ(κ1,κ2)K(x)κ2dqx−K(κ2+μ(κ1,κ2))+qK(κ2)[2]q|≤qμ(κ1,κ2)[ρ2(q)|κ2DqK(κ1)|+ρ1(q)|κ2DqK(κ2)|−χμ(κ1,κ2)θρ4(q)], | (3.37) |
where ρ1(q), ρ2(q) and ρ4(q) are defined in Theorem 3.3.
Theorem 3.7. If we assume all the conditions of Lemma 3.2, then the following inequality, shows that |κ2DqK|σ is generalized higher-order strongly preinvex function of order θ>0 with modulus χ≥0 on [κ2+μ(κ1,κ2),κ2] for 1p+1σ=1, then
|1μ(κ1,κ2)∫κ2κ2+μ(κ1,κ2)K(x)κ2dqx−K(κ2+μ(κ1,κ2))+qK(κ2)[2]q|≤qμ(κ1,κ2)[ρ3(p,q)]1p×(|κ2DqK(κ1)|σ+q|κ2DqK(κ2)|σ[2]q−χμ(κ1,κ2)θq2[2]q[3]q)1σ, | (3.38) |
where ρ3(p,q) is defined in Theorem 3.4.
Proof. The desired inequality (3.38) can be obtained by applying the strategy used in the proof of Theorem 3.4 and taking into account the Lemma 3.8.
Theorem 3.8. If we assume all the conditions of Lemma 3.2, then the following inequality, shows that |κ2DqK|σ is generalized higher-order strongly quasi-preinvex function of order θ>0 with modulus χ≥0 on [κ2+μ(κ1,κ2),κ2] for σ≥1, then
|1μ(κ1,κ2)∫κ2κ2+μ(κ1,κ2)K(x)κ2dqx−K(κ2+μ(κ1,κ2))+qK(κ2)[2]q|≤qμ(κ2,κ1)(ρ3(q))1−1σ(ρ3(q)ρ9(q)−χμ(κ1,κ2)θρ4(q))1σ, | (3.39) |
where ρ9(q)=max{|κ2DqK(κ1)|σ,|κ2DqK(κ2)|σ}.
Proof. The desired inequality (3.39) can be obtained by applying the strategy used in the proof of Theorem 3.5 and taking into account the Lemma 3.8.
In this section we compare our results with the existing results graphically.
Consider the function K:[0,3]→R defined by K(ω)=ω2. Then K is a continuous function on [0,3]⊂R and is qκ1-differentiable on [0,3]. Its qκ1-derivative at ω is given by
0DqK(ω)=[2]qω,ω≠0 |
which is continuous and qκ1-integrable on [0,3] for q∈(0,1).
Let σ=4, χ=2=θ, and μ(3,0)=3, hence
|0DqK(ω)|4=[2]4qω4,ω≠0. |
We observe that
|0DqK(0)|4=limx→0+[2]4qω4=0 |
and
|0DqK(3)|4=81[2]4q. |
The RHS of the inequality (2.9) becomes
α(q)=9q[2]q(q(2+q+q3)[2]3q)34(q[2]q(1+3q2+2q3)[3]q)14 | (4.1) |
and the RHS of the inequality (3.7) takes the form
β(q)=3q(2q[2]q(q2+q+1))34(q81[2]4qq4+q3+2q2+q+1−18q2(q4+q3+q2−q+1)q9+3q8+6q7+9q6+11q5+11q4+9q3+6q2+3q+1)14. | (4.2) |
From Figure 1, it can be seen that β(q)>α(q), i.e. the inequality (2.9) provides better estimate than that of the inequality (3.9).
Now consider the RHS of the inequality (2.10)
γ(q)=9q2(2+q+q3)[2]3q | (4.3) |
and the RHS of the inequality (3.29)
δ(q)=3q(2qq3+2q2+2q+1)34(162q[2]4qq3+2q2+2q+1−18q2(q4+q3+q2−q+1)q9+3q8+6q7+9q6+11q5+11q4+9q3+6q2+3q+1)14. | (4.4) |
From Figure 2, it can be seen that δ(q)>γ(q), i.e. the inequality (2.10) provides better estimate than that of the inequality (3.29).
In this research, the generalized class of preinvex functions has been considered. We also obtained attractive quantum analogs of new Hermite-Hadamard type inequalities for generalized higher-order strongly preinvex and quasi-preinvex functions. New integral identities for qκ1- and qκ2-differentiable functions were proven, which played an important part in obtaining quantum estimates of Hermite-Hadamard type inequalities for qκ1- and qκ2-differentiable generalized higher-order strongly preinvex and quasi-preinvex functions. Our study's claim has been graphically supported. Finally, the innovative definition of generalized higher-order strongly preinvex functions has potential applications in parallelogram law of Lp-spaces in functional analysis and opening new avenues for future study. Moreover, Srivastava [19] we presented (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. There is also a clear connection between the classical q-analysis, which we used here, and the so-called (p,q)-analysis. We emphasize that the results for the q-analogues, which we discussed in this article for 0<q<1, can be easily (and probably trivially) converted into the corresponding results for the (p,q)-analogues (with 0<q<p≤1) by making a few obvious parametric and argument changes, with the additional parameter p being superfluous.
The authors are very grateful to the anonymous referees, for several valuable and helpful comments, suggestions and questions, which helped them to improve the paper into present form.
The work was supported by Zhejiang Normal University.
The authors declare that they have no competing interests.
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