Generalizations on Some Hermite-Hadamard Type Inequalities for Differentiable Convex Functions with Applications to Weighted Means

Some new Hermite-Hadamard type inequalities for differentiable convex functions were presented by Xi and Qi. In this paper, we present new generalizations on the Xi-Qi inequalities.

In this paper, we generalize the Xi-Qi inequalities.

Main Results
Proof. Suppose that | ( )| is convex on [ , ] with ≥ 1. By Lemma 1, we have 4 The Scientific World Journal Case ( = 1). By the convexity of | ( )| and Lemma 2, we have Thus, Case ( > 1). By Hölder's inequality, we have The Scientific World Journal By the convexity of | ( )| and Lemma 2, we have Thus, This proof is completed.
It is easy to notice that if we put = ( − )/2 in Theorem 3 then we get the following.
One can easily check that if we put = 2( − )/3 in Theorem 3 then we get the following.
It is easy to notice that if we put = = 0 in Theorem 3 then we get the following.
The Scientific World Journal 7 It is easy to notice that if we put = = 1/2 in Theorem 3 then we get the following.
It is easy to notice that if we put = = 1 in Theorem 3 then we get the following.
The Scientific World Journal Next, we suppose that > 1. By Lemma 1 and Hölder's inequality, we have By the convexity of | ( )| and Lemma 2, we have Thus, This proof is completed.
It is easy to notice that if we put = ( − )/2 in Theorem 10 then we get the following.
Corollary 11 (see [8] One can easily check that if we put = ( − )/3 in Theorem 10 then we get the following.
The Scientific World Journal One can easily check that if we put = 2( − )/3 in Theorem 10 then we get the following.
It is easy to notice that if we put = = 0 in Theorem 10 then we get the following.
It is easy to notice that if we put = = 1/2 in Theorem 10 then we get the following. ( It is easy to notice that if we put = = 1 in Theorem 10 then we get the following.

Applications
In this section, we suppose that The generalized logarithmic mean of data { , } is defined by The identric mean of data { , } is defined by Applying Corollary 7 with ( ) = on (0, ∞), we get the following: Applying Corollary 9 with ( ) = on (0, ∞), we get the following: Applying Corollary 7 with ( ) = ln on (0, ∞), we get the following: Applying Corollary 9 with ( ) = ln on (0, ∞), we get the following:

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.