Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping. Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–4]). A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5]. Let us consider the bidimensional interval Δ := [a, b] × [c, d] in R with a < b and c < d; a mapping f : Δ → R is said to be convex on Δ if the inequality


Introduction
Let : ⊆ R → R be a convex function and , ∈ with < ; we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping. Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1][2][3][4]).
A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].
A formal definition for coordinated convex functions may be stated as follows.
Dragomir in [5] established the following Hadamardtype inequalities for coordinated convex functions in a rectangle from the plane R 2 .
In ([9], 2012), Latif and Dragomir obtained some new Hadamard type inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables based on the following lemma.
In ( [10], 2012), analogous results which are related to the right-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables were proved by Sarıkaya et al. based on the following lemma.
Theorem 10 (see [10]). Let : V| is convex on the coordinates on Δ and ≥ 1, then the following equality holds: where is as given in Theorem 8.
In [11], Ozdemir et al. established some Simpson's inequalities for coordinated convex functions based on the following lemma.
In this paper, a generalized lemma is proved and several new inequalities for differentiable coordinated convex and concave functions in two variables are obtained.

Lemmas
To establish our results, we need the following lemma.
, then the following equality holds: Chinese Journal of Mathematics Proof. Since thus, by integration by parts, it follows that Similarly, we can get Now Multiplying both sides by ( − )( − ) and using the change of the variable = + (1 − ) and = V + (1 − V) , which completes the proof.

Main Results
Theorem 15. Let : Δ ⊆ R 2 → R be a partial differentiable mapping on Δ := [ , ] × [ , ] in R 2 with < and < . If | 2 / V| is convex on the coordinates on Δ and ∈ [0, 1], then the following equality holds: Because | 2 / V| is a convex function on the coordinates on Δ, then one has On the other hand, we have which completes the proof.
Proof. From Lemma 13, we obtain By using the well-known Hölder inequality for double integrals, then one has Because | 2 / V| is a convex function on the coordinates on Δ, by (4), then one has We note that Hence, it follows that Remark 18. Applying Theorem 17 for = 0, 1, we get the results of Theorems 5 and 9, respectively.
Proof. From Lemma 13, we obtain By using the well-known power mean inequality for double integrals, then one has Because | 2 / V| is a convex function on the coordinates on Δ, then one has Thus, it follows that On the other hand, we obtain Thus, we get the following inequality: which completes the proof.
Proof. Similarly as in Theorem 17, because | 2 / V| is a concave function on the coordinates on Δ, by the reversed direction of (4), we get which yields the desired result.

Conflict of Interests
The author has declared that no conflict of interests exists.