On New Estimates of q -Hermite–Hadamard Inequalities with Applications in Quantum Calculus

: In this paper, we ﬁrst establish two quantum integral ( q -integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q -midpoint and q - trapezoidal estimates for the newly established q -Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q -differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities.


Introduction
In recent studies, fractional calculus has proved to be the among of the most widely used areas of mathematical science.This is because, we can see the activities of researchers in this field.Besides, there have been published papers in which fixed point theorems play a key role in existence results for given fractional differential equations [1][2][3].Due to the expansion of this branch of mathematics, mathematicians studied a new field in which the concept of limit has no role in the definitions of operators.Also, because of the fundamental role of the quantum parameter q, they called it the theory of quantum fractional calculus.The initial steps in this field were taken by Jackson [4,5] and then, it was extended to more practical fields such as combinatorics, quantum mechanics, discrete mathematics, hypergeometric series, particle physics, and theory of relativity.To remember and fully understand the concepts of q-calculus, one can mention the sources [6][7][8].
Consider the function ρ : I → R so that I is a real interval.Then, ρ is called a convex function if holds for each t ∈ [0, 1] and x, y ∈ I.
From [20], it is established that ρ is convex if and only if ρ satisfies the Hermite-Hadamard inequality, formulated as for each ν, σ ∈ I with ν < σ.
Motivated by the ongoing research, we obtain another version of q-Hermite-Hadamard inequality in consideration of convex mappings, and prove some new q-midpoint type inequalities for convex mappings of the q-differentiable type.Also, in some examples, we show that the newly obtained inequalities are the generalizations of the existing Hermite-Hadamard inequality and midpoint inequalities.These new results can be used for finding some error bounds for the midpoint and trapezoidal rules in q-integration formulas that are very important in the field of numerical analysis.
This paper is organized as follows: The basics of quantum calculus along with other topics in the present area are addressed briefly in the next section.In Sections 3 and 4, some q-midpoint and q-trapezoid type estimates are studied for the inequality (4) under the q-differentiable functions.The connection between our results and other results in the literature are also stated.We provide some mathematical examples in Section 5 to demonstrate the validity of the newly developed inequalities.Section 6 concludes the paper by giving some ideas for the future.

Preliminaries of q-Calculus
In the preliminaries, we collect the definitions and several properties of quantum operators.Along with these, some famous inequalities are restated with respect to quantum integrals.In the whole of the article, 0 < q < 1 is constant.
If ν = 0, then (9) becomes It is the same q-Jackson integral [4,38,39].Later, Bermudo et al. extended the following new quantum operators, which are introduced as the right q-operators.

q-Trapezoidal Inequalities
In this section, we establish some right estimates of the inequality (4) using differentiable convex functions.
Proof.By Lemma 2, we compute Similarly, from Lemma 1, we get Thus, we obtain the desired identity by combining (11) and (12).
Theorem 1.Under the hypotheses of Lemma 3, we have the following inequality if σ D q ρ and ν D q ρ are convex: Proof.From Lemma 3 and using the convexity of ν D q ρ and σ D q ρ , we obtain which completes the proof.
Theorem 2. Under all the hypotheses of Lemma 3, we have the following inequality if σ D q ρ p 1 and ν D q ρ p 1 , p 1 ≥ 1 are convex: Proof.The power mean inequality and Lemma 3 give By the convexity of ν D q ρ p 1 and σ D q ρ p 1 , we have Thus, the proof is completed.
Theorem 3.Under the hypotheses of Lemma 3, the following inequality is satisfied if σ D q ρ p 1 and ν D q ρ p 1 , p 1 > 1 are convex: Proof.The Hölder inequality and Lemma 3 give By the convexity of ν D q ρ p 1 and σ D q ρ p 1 , we have Thus, the proof is completed.

q-Midpoint Inequalities
In this section, we establish some right estimates of inequality (4) for differentiable convex functions.
such that ν D q ρ and σ D q ρ are integrable and continuous on [ν, σ], then Proof.It can be easily proved by following the procedure used in Lemma 3.
Theorem 4.Under the hypotheses of Lemma 4, the following inequality holds if σ D q ρ and ν D q ρ are convex: + σ D q ρ(ν) 3 Proof.It can be easily proved by following the procedure used in Theorem 1.
Theorem 5.Under the hypotheses of Lemma 4, this inequality is satisfied if σ D q ρ p 1 and ν D q ρ p 1 , p 1 ≥ 1 are convex: Proof.It can be easily proved by following the procedure used in Theorem 2.
Theorem 6.Under the hypotheses of Lemma 4, we have the following inequality if σ D q ρ p 1 and Proof.It can be easily proved by following the procedure used in Theorem 3.

Examples
In this section, we show the validity of the established inequalities using some examples.
Example 2. For a convex function ρ : [0, 1] → R given by ρ(x) = x 2 + 2, by ( 14) with q = 1 2 and p 1 = 2, the left side of the inequality and the right side of it becomes It is clear that 0.09 < 0.35.
Example 3.For a convex function ρ : [0, 1] → R given by ρ(x) = x 2 + 2, from ( 16) with q = 1 2 and p 1 = r 1 = 2, the left side of the inequality and the right side of it becomes It is clear that 0.09 < 0.45.
Example 5.For a convex function ρ : [0, 1] → R given by ρ(x) = x 2 + 2, by ( 18) with q = 1 2 , the left side of the inequality and the right side of it becomes where p 1 = 2, it is clear that 0.15 < 0.50.

Conclusions
In this paper, new variants of midpoint and trapezoidal inequalities for differentiable convex functions in the framework of q-calculus are established.We also used well-known power mean and Hölder inequalities to find q-type of trapezoidal and midpoint inequalities in consideration of q-differentiable convex mappings.These new results can be used for finding some error bounds for the midpoint and trapezoidal rules in q-integration formulas that are very important in the field of numerical analysis.It is an interesting idea that other mathematicians in this field can derive new inequalities for quantum coordinated convex mappings.