Keywords: generalization, HermiteHadamard integral inequality, differentiable function, Hölder integral inequality
Turkish Journal of Analysis and Number Theory, 2015 3 (2),
pp 4348.
DOI: 10.12691/tjant322
Received March 02, 2015; Revised April 10, 2015; Accepted April 17, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
Let f(x) be a convex function on [a; b], the famous HermiteHadamard integral inequality may be expressed as
 (1.1) 
It is well known that HermiteHadamard integral inequality is an important cornerstone in mathematical analysis and optimization. There has been a growing literature considering its refinements and interpolations. For more information, please refer to the monographs ^{[3, 4]}, the newly published papers ^{[1, 7]}, and plenty of references therein.
The following theorems are some refinements and generalizations of inequalities in (1.1).
Theorem 1.1 (^{[2]} and [^{[5]}, Theorem A]). Let be a twice differentiable mapping and suppose that for all Then we have
 (1.2) 
and
 (1.3) 
This theorem was generalized as follows.
Theorem 1.2 (^{[6]} and [^{[5]}, Theorem B]). Let be a twice differentiable mapping and suppose that for all , then
 (1.4) 
and
 (1.5) 
where
 (1.6) 
The above two theorems were further generalized by the following theorems.
Theorem 1.3 ([^{[5]}, Theorem 1]). Let be ntime differentiable on the closed interval such that for and . Further, let be a parameter. Then
 (1.7) 
where is defined by (1.6).
Theorem 1.4 ([^{[5]}, Theorem 3]). Let and f(t) be ntime differentiable on the closed interval such that for and Then
 (1.8) 
where is defined by (1.6).
Theorem 1.5 ([^{[5]}, Theorem 5]). Let be a harmonic sequence of polynomials, that is,
 (1.9) 
and for all defined and . Further let be time differentiable on such that for and . Then, for any constant , we have
 (1.10) 
and
 (1.11) 
where is defined by (1.6).
Theorem 1.6 ([^{[5]}, Theorem 7]). Let and be two harmonic sequences of polynomials, and be two real constants, and . Further let be time differentiable on such that for and . Then
 (1.12) 
and
 (1.13) 
Where is defined by (1.6) and
The aim of this paper is to, by establishing two integral identities and Hölder integral inequality, generalize the above six theorems recited from ^{[5]} to more general cases.
2. Lemmas
For generalizing the above six theorems recited from ^{[5]} to more general cases, we need the following integral identities.
Lemma 2.1. For , let be a time differentiable function on , and let and be time differentiable functions for some, than
 (2.1) 
where
 (2.2) 
and for .
Proof. When , it is not difficult to obtain that
Suppose that the inequality (2.1) holds for . For , by integration by parts, we obtain
By induction, the proof of inequality (2.1) is complete.
Lemma 2.2 For , let be a time differentiable function on and, for let and be time differentiable functions, then
 (2.3) 
where
 (2.4) 
and for are same with (2.2).
3. Main results
Now we are in a position to generalize the above six theorems recited from ^{[5]} to more general cases.
Theorem 3.1. For , let be time differentiable such that for . for let are time differentiable functions. Then
 (3.1) 
where is defined by (1.6), and are defined as in (2.2) and
 (3.2) 
Proof. By Lemma 2.1, we have
 (3.3) 
and
 (3.4) 
On the other hand, by the Hölder inequality,
 (3.5) 
and
 (3.6) 
Combining (3.3) to (3.6) yields (3.1). Theorem 3.1 is thus proved.
Remark 1. From taking
in (3.1), the double inequality (1.7) followes.
If taking in Theorem 3.1, we can derive the following corollary.
Corollary 3.1.1. For , let be time differentiable such that for and let betime differentiable. Then
 (3.7) 
Proof. This follow from putting , and in Theorem 3.1.
Remark 2. If letting for in (3.7), the double inequality (1.8) may be derived.
Corollary 3.1.2. Under the conditions of Theorem 3.1, if , then
 (3.8) 
Corollary 3.1.3. Under the conditions of Theorem 3.1, if , then
 (3.9) 
Theorem 3.2. For , let be a time differentiable function on and, for let and be time differentiable functions. Then, for being real constants,
 (3.10) 
and
 (3.11) 
where and are defined respectively by (1.6), (3.2), (2.4), (2.2) and
 (3.12) 
Proof. Applying Lemma 2.2 results in
and
It is easy to show, by the Hölder inequality, that
and
Combining the above identitie and inequalities yields Theorem 3.2.
Remark 3.3. For , setting
in Theorem 3.2, where and are two harmonic sequences of polynomials, reveals the double inequalities (1.12) and (1.13).
Corollary 3.1.1. Let be time differentiable on the closed interval such that for , and be time differentiable function for , let be a real constant. Then
 (3.13) 
and
 (3.14) 
Proof. This follows from taking and in Theorem 3.2.
Remark 3.4. Taking in (3.13) and (3.14), be a harmonic of polynomials may derive the double inequalities (1.10) and (1.11).
Corollary 3.2.2. Under the conditions of Theorem 3.2, we have
 (3.15) 
and
 (3.16) 
Proof. This follows from putting in Theorem 3.2.
Corollary 3.2.3. Under the conditions of Theorem 3.2, if , then
 (3.17) 
and
 (3.18) 
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China underGrant No. 11361038, China and by the Foundationof the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY14191, China.
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