Some New Quantum Hermite-Hadamard Type Inequalities for s -Convex Functions

: In this investigation, we ﬁrst establish new quantum Hermite–Hadamard type integral inequalities for s -convex functions by utilizing newly deﬁned T q -integrals. Then, by using obtained inequality, we establish a new Hermite–Hadamard inequality for coordinated ( s 1 , s 2 ) -convex functions. The results obtained in this paper provide signiﬁcant extensions of other related results given in the literature. Finally, some examples are given to illustrate the result obtained in this paper. These types of analytical inequalities, as well as solutions, apply to different areas where the concept of symmetry is important.


Introduction
The Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [1,2], p. 137) is one of the most well established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that if F : I → R is a convex function on the interval I of real numbers and σ, ρ ∈ I with σ < ρ, then Over the years, a large number of studies have focused on finding trapezoid and midpoint type inequalities that provide boundaries to the right and left sides of inequality (1), respectively. Dragomir and Agarwal first considered the trapezoid inequalities for differentiable convex mappings in [3] whereas Kirmacı first proved the midpoint inequalities for differentiable convex mappings in the paper [4]. In [5], Sarikaya et al. extended the inequalities (1) to the case of Riemann-Liouville fractional integrals and the authors also established some corresponding fractional trapezoid type inequalities. What's more, Sarikaya obtained fractional Hermite-Hadamard inequalities and fractional trapezoids in the case of the functions with two variables in [6]. In [7], Dragomir proved Hermite-Hadamard type inequalities for coordinated convex functions R 2 . For some other related papers on Hermite-Hadamard type inequalities for convex functions and other kinds of convex classes, please refer to [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].
Dragomir and Fitzpatrick [29] used this class of functions and proved the following Hermite-Hadamard inequality: Definition 2. A function F : ∆ ⊂ R 2 → R will be called coordinated (s 1 , s 2 )-convex functions on ∆, if the following inequality Here, if we put s 1 = 1 and s 2 = 1 , then coordinated (s 1 , s 2 )-convexity reduces to coordinated convexity.

Quantum Calculus
In this section, we summarize some required definitions of quantum calculus and important quantum integral inequalities. For for information about the related results, one can refer to the papers [37][38][39][40][41][42][43][44] 2.1. q-Integrals and Related Inequalities Set the following notation (see [38]): The [n] q is set of integers and expressed as [n] q = 1−q n 1−q = 1 + q + q 2 + . . .. . .q n−1 with q ∈ (0, 1). Jackson derived the q-Jackson integral in [37] from 0 to ρ for q ∈ (0, 1) as follows: provided the sum converges absolutely. The q-Jackson integral in a generic interval [σ, ρ] was given by in [37] and defined as follows: The quantum integrals on the interval [σ, ρ] is defined as follows: In [45], Alp et al. proved the corresponding Hermite-Hadamard inequalities for convex functions by using q σ -integrals, as follows: Theorem 1. If F : [σ, ρ] → R be a convex differentiable function on [σ, ρ] and 0 < q < 1. Then, q-Hermite-Hadamard inequalities Bermudo et al. proved the corresponding Hermite-Hadamard inequalities for convex functions by using q ρ -integrals, as follows: Then, q-Hermite-Hadamard inequalities From Theorems 1 and 2, one can write the following inequalities: 46]). For any convex function F : [σ, ρ] → R and 0 < q < 1, we have and In [47], Latif defined q σ -integral for functions of two variables and presented important properties of this integral. In [48], Alp and Sarikaya proved quantum Hermite-Hadamard inequalities for co-ordinated convex functions. On the other hand, Budak et al. [49] defined the q d σ , q ρ and q ρd -integrals for functions of two variables and they also gave the corresponding Hermite-Hadamard inequalities for these newly defined integrals.

T q -Integrals and T q -Hermite-Hadamard Inequalities
In this subsection, we summarize the definitions and some properties of the T q -Integrals. Alp and Sarikaya defined the following new version of quantum integral, which is called σ T q -integral.
In [43], Kara et al. introduced the following generalized quantum integral, which is called ρ T q -integral.
Kara and Budak defined T q -integrals for two-variables functions, as follows and respectively.
Kara and Budak proved the corresponding Hermite-Hadamard type inequalities for these T q -integrals. In this paper, we generalize the results proved in the papers [42][43][44] for s-convex functions.

Generalized Quantum Hermite Hadamard Inequalities
In this section, we establish new T q Hermite-Hadamard type integral inequalities for s-convex and coordinated (s 1 , s 2 )-convex functions.
To prove the second inequality, we use the s-convexity, we have Thus, By taking σ T q -integral of (19) on [0, 1] and by using Definitions 5 and 6, we have Thus, the proof is accomplished.

Remark 1.
In Theorem 5, if we take the limit as q → 1, then the inequality (17) becomes the inequality (3).

Applications
In this section, we give some applications to illustrate our main outcomes.

Conclusions
In current work, some new quantum Hermite-Hadamard-type inequalities for sconvex and coordinated (s 1 , s 2 )-convex functions by utilizing the T q -integrals are obtained. We also presented some examples which satisfied our main outcomes. It is also shown that some classical results can be obtained by the results presented in the current study by taking the limit q → 1. It is a novel and fascinating problem that the researcher will be able to obtain similar inequalities in their future work for various types of convexity and coordinated convexity.