Abstract
Keywords:
HermiteHadamard’s integral inequality; differentiable function; convex function1 Introduction
Throughout this paper, we adopt the following notations:
We recall some definitions of several convex functions.
Definition 1.1 A function is said to be convex if
Definition 1.2 ([1])
is valid for all and , then we say that is an mconvex function on .
Definition 1.3 ([2])
is valid for all and , then we say that is an convex function on .
In recent decades, plenty of inequalities of HermiteHadamard type for various kinds of convex functions have been established. Some of them may be reformulated as follows.
Theorem 1.1 ([[3], Theorem 2.2])
Letbe a differentiable mapping andwith. Ifis convex on, then
Theorem 1.2 ([[4], Theorem 2])
Letbemconvex and. Iffor, then
Theorem 1.3 ([[2], Theorem 2.2])
Letbe an open interval and letbe a differentiable function such thatfor. Ifismconvex onfor someand, then
Theorem 1.4 ([[2], Theorem 3.1])
Letbe an open interval and letbe a differentiable function such thatfor. Ifisconvex onfor someand, then
where
and
For more and detailed information on this topic, please refer to the monograph [5] and newly published papers [616].
In this paper, we establish some HermiteHadamard type integral inequalities for ntime differentiable functions which are convex.
2 A lemma
In order to find inequalities of HermiteHadamard type for convex functions, we need the following lemma.
Lemma 2.1 ([[17], Lemma 2.1] or [[18], Lemma 2.1])
Letbe anntime differentiable function such thatforis absolutely continuous on. Then the identity
holds for all, where the kernelis defined by
3 HermiteHadamard type inequalities for convex functions
We now set off to establish some new integral inequalities of HermiteHadamard type for ntime differentiable convex functions.
Theorem 3.1Letbe anntime differentiable function forand letand. Ifandforisconvex on, then
Proof If , by Lemma 2.1, Hölder’s integral inequality, and the convexity of , we have
Substituting
and
into the above inequality leads to the inequality (3.1) for .
If or , by virtue of Lemma 2.1 and the property that is convex on , we have
and
The inequality (3.1) for or follows. Theorem 3.1 is thus proved. □
Corollary 3.1Under the conditions of Theorem 3.1,
Corollary 3.2Under the conditions of Theorem 3.1,
Theorem 3.2Letandbe anntime differentiable function for, and letand. If, forisconvex on, and, then
Proof When , by Lemma 2.1 and Hölder’s integral inequality, we have
where
and
and
Hence, the inequality (3.4) follows.
When or , the proof of the inequality (3.4) is similar to the above argument. The proof of Theorem 3.2 is complete. □
Corollary 3.3Under the conditions of Theorem 3.2,
Corollary 3.4Under the conditions of Theorem 3.2,
Corollary 3.5Under the conditions of Theorem 3.2,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
This work was supported partially by Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103 and the National Natural Science Foundation of China under Grant No. 10962004.
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Xi, BY, Wang, SH, Qi, F: Some inequalities of HermiteHadamard type for functions whose 3rd derivatives are Pconvex . Appl. Math.. 3(12), 1898–1902 (2012) Available online at http://dx.doi.org/10.4236/am.2012.312260
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