On the Hermite-Hadamard Type Inequalities

In the present paper, we establish some new Hermite-Hadamard type inequalities involving two functions. Our results in a special case yield recent results on Hermite-Hadamard type inequalities.


Introduction
The following inequality is well known in the literature as Hermite-Hadamard's inequality [].

Theorem . Let f : [a, b] ⊂ R → R be a convex function on an interval of real numbers.
Then the following Hermite-Hadamard inequality for convex functions holds: If the function f is concave, the inequality (.) can be written as follows: Recently, many generalizations, extensions and variants of this inequality have appeared in the literature (see, e.g., [-]) and the references given therein. In particular, in , Özdemir and Dragomir [] established some new Hermite-Hadamard inequalities and other integral inequalities involving two functions in R. Following this work, the main purpose of the present paper is to establish some dual Hermite-Hadamard type inequalities involving two functions in R  . Our results provide some new estimates on such type of inequalities.

Preliminaries
A region D ⊂ R  is called convex if it contains the close line segment joining any two of its points, or equivalently, if λx  + ( -λ)x  , λy  + ( -λ)y  ∈ D whenever x(x  , y  ), y(x  , y  ) ∈ D and  ≤ λ ≤ . http://www.journalofinequalitiesandapplications.com/content/2013/1/228 Let z = f (x, y) be a duality function on the convex region D ⊂ R  . z = f (x, y) is called a duality convex function on the convex region D if whenever (x  , y  ), (x  , y  ) ∈ D and  ≤ λ ≤ .

Main results
Our main results are established in the following theorems.

where B(p, q) is the Barnes-Godunova-Levin constant given by (.).
Proof Observe that whenever . Namely, that is, and p > , using the power-mean inequality (.), we obtain .
In view that f p (x, y) and g q (x, y) are concave functions on [a, b] × [c, d], from Lemma ., we get (  .  ) http://www.journalofinequalitiesandapplications.com/content/2013/1/228 By multiplying the above inequalities, we obtain If p, q > , then it is easy to show that Thus, by applying Barnes-Godunova-Levin inequality to the right-hand side of (.) with (.), (.), we get (.). The proof is complete.