Weighted Ostrowski type inequalities for co-ordinated convex functions

In this paper, utilizing an identity given by Yıldız and Sarıkaya in (Yildiz and Sarikaya in Int. J. Anal. Appl. 13(1):64–69, 2017), we establish some weighted Ostrowski type inequalities for co-ordinated convex functions in a rectangle from the plane R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R} ^{2}$\end{document}. Moreover, as special cases of our main results, we give some weighted Hermite–Hadamard type inequalities. The results given in this paper provide generalizations of some result established in earlier works.


Introduction
In the history of calculus development, integral inequalities have been thought of as a key factor in the theory of differential and integral equations. The study of various types of integral inequalities has been in the focus of great attention of a number of scientists interested in both pure and applied mathematics for more than a century. One of the many fundamental mathematical discoveries of A. M. Ostrowski [15] is the following classical integral inequality associated with the differentiable mappings: Let F : [ρ 1 , ρ 2 ]→ R be a differentiable mapping on (ρ 1 , ρ 2 ) whose derivative F : (ρ 1 , ρ 2 )→ R is bounded on (ρ 1 , ρ 2 ), i.e., F ∞ = sup ψ∈(ρ 1 ,ρ 2 ) |F (ψ)| < ∞. Then, the inequality holds: for all κ ∈ [ρ 1 , ρ 2 ]. The constant 1 4 is the best possible. The other important fundamental result, the Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [7], [18, p. 137]), is one of the most wellestablished 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 ρ 1 , ρ 2 ∈ I with ρ 1 < ρ 2 , then Both inequalities hold in the reversed direction if F is concave. We note that the Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity, and it follows easily from Jensen's inequality. The Hermite-Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied. A formal definition for co-ordinated convex function may be stated as follows: 1], if it satisfies the following inequality: The mapping F is a co-ordinated concave on if inequality (1.1) holds in reversed direction for all ψ, ϕ ∈ [0, 1] and (κ, u), (γ , v) ∈ .
Barnet and Dragomir gave the following Ostrowski type inequalities for double integrals in [5].
Theorem 2 Suppose that F : → R is co-ordinated convex, then we have the following inequalities: The above inequalities are sharp. The inequalities in (1.3) hold in the reverse direction if the mapping F is a co-ordinated concave mapping.
Over the years, many papers have been dedicated to the generalizations and new versions of the inequalities (1.2) and (1.3) using the different types of convex functions. For the other Ostrowski and Hermite-Hadamard type inequalities for co-ordinated convex functions, please refer to ( [1-4, 8-14, 16, 17, 19-29]) In [30], Yıldız and Sarıkaya proved the following Lemma.
be an integrable function on and let F : → R be an absolutely continuous function such that the partial derivatives of order ∂ψ∂ϕ exist for all (ψ, ϕ) ∈ . Then we have the identity The aim of this paper is to establish some weighted generalizations of the Ostrowski type integral inequalities. The results presented in this paper provide extensions of those given in [14].

Weighted Ostrowski type inequalities
In this section, using Lemma 1, we established some weighted Ostrowski type inequalities for co-ordinated convex mapping.
First, we define the following mapping Using the change of variables in Lemma 1, we have the following identity:

Theorem 3 Suppose that the mapping w is as in Lemma
∂ψ∂ϕ | is a co-ordinated convex function on , then for all (κ, γ ) ∈ we have the following inequality where the mapping is defined as in (2.1).
Proof By taking the modulus of the equality (2.2), we have Since w(κ, γ ) is bounded on , and | ∂ 2 F ∂ψ∂ϕ | is co-ordinated convex on , we obtain Similarly, we have This completes the proof.

Theorem 4 Let w be as in Theorem
∂ψ∂ϕ | q is a co-ordinated convex function on , then for all (κ, γ ) ∈ , we have the following inequality where the mapping is defined as in (2.1), and 1 p + 1 q = 1.