On some Hermite–Hadamard type inequalities for (s, QC)-convex functions

In the paper, the authors introduce a new notion “\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s,\text {QC})$$\end{document}(s,QC)-convex function on the co-ordinates” and establish some Hermite–Hadamard type integral inequalities for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s,\text {QC})$$\end{document}(s,QC)-convex functions on the co-ordinates.

Definition 4 (Xi and Qi 2015a) For some s ∈ [−1, 1], a function f : I ⊆ R → R is said to be extended s-convex if is valid for all x, y ∈ I and ∈ (0, 1).
Definition 5 (Dragomir 2001;Dragomir and Pearce 2000) A function f : = [a, b] × [c, d] ⊆ R 2 → R is said to be convex on co-ordinates on if the partial functions are convex for all x ∈ (a, b) and y ∈ (c, d).
Remark 1 By Definitions 8 and 10 and Lemma 1, we see that, for s ∈ [−1, 1] and f : ⊆ R 2 → R 0 , 1. If f : → R 0 is a J-quasi-convex function on the co-ordinates on , then f is a (Js, JQC)-convex function on the co-ordinates on ; 2. Every J-quasi-convex function f : → R 0 is a (Js, JQC)-convex function on the coordinates on .
Remark 3 Considering Definitions 9 and 12 and Lemma 1, for s ∈ [−1, 1] and f : ⊆ R 2 → R 0 , 1. If f : → R 0 is a quasi-convex function on the co-ordinates on , then it is an (s, QC)-convex function on the co-ordinates on ; 2. Every quasi-convex function f : → R 0 is an (s, QC)-convex function on the coordinates on .
Proof This follows from a straightforward computation.

Some integral inequalities of Hermite-Hadamard type
In this section, we will establish Hermite-Hadamard type integral inequalities for (s, QC) -convex functions on the co-ordinates on rectangle from the plane R 2 .
∂x∂y q is an (s, QC)-convex function on the co-ordinates on with a < b and c < d for some s ∈ [−1, 1] and q ≥ 1, then When s = −1, similar to the proof of inequalities (21) to (24), we can write Substituting inequalities (25) to (28) into (20) leads to the inequality (18). Theorem 4 is thus proved.