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Some Generalizations of Integral Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions

Tian-Yu Zhang, Bai-Ni Guo*

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

Abstract: In the paper, by establishing two integral identities and Hölder integral inequality, the authors generalize some integral inequalities of Hermite-Hadamard type for n-time differentiable functions on a closed interval.

Keywords: generalization, Hermite-Hadamard integral inequality, differentiable function, Hölder integral inequality

Received April 02, 2015; Revised April 12, 2015; Accepted April 17, 2015

Citation: Tian-Yu Zhang, Bai-Ni Guo, Some Generalizations of Integral Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions. Turkish Journal of Analysis and Number Theory, 3 (2). 43-48. January 2015. doi:10.12691/tjant-3-2-2

1. Introduction

Let f(x) be a convex function on [a; b], the famous Hermite-Hadamard integral inequality may be expressed as

[picture] | (1.1)

It is well known that Hermite-Hadamard 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

[picture] | (1.2)

and

[picture] | (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

[picture] | (1.4)

and

[picture] | (1.5)

where

[picture] | (1.6)

The above two theorems were further generalized by the following theorems.

Theorem 1.3 ([[5], Theorem 1]). Let be n-time differentiable on the closed interval such that for and . Further, let be a parameter. Then

[picture] | (1.7)

where is defined by (1.6).

Theorem 1.4 ([[5], Theorem 3]). Let and f(t) be n-time differentiable on the closed interval such that for and Then

[picture] | (1.8)

where is defined by (1.6).

Theorem 1.5 ([[5], Theorem 5]). Let be a harmonic sequence of polynomials, that is,

[picture] | (1.9)

and for all defined and . Further let be -time differentiable on such that for and . Then, for any constant , we have

[picture] | (1.10)

and

[picture] | (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

[picture] | (1.12)

and

[picture] | (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

[picture] | (2.1)

where

[picture] | (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

[picture] | (2.3)

where

[picture] | (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

[picture] | (3.1)

where is defined by (1.6), and are defined as in (2.2) and

[picture] | (3.2)

Proof. By Lemma 2.1, we have

[picture] | (3.3)

and

[picture] | (3.4)

On the other hand, by the Hölder inequality,

[picture] | (3.5)

and

[picture] | (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 be-time differentiable. Then

[picture] | (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

[picture] | (3.8)

Corollary 3.1.3. Under the conditions of Theorem 3.1, if , then

[picture] | (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,

[picture] | (3.10)

and

[picture] | (3.11)

where and are defined respectively by (1.6), (3.2), (2.4), (2.2) and

[picture] | (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

[picture] | (3.13)

and

[picture] | (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

[picture] | (3.15)

and

[picture] | (3.16)

Proof. This follows from putting in Theorem 3.2.

Corollary 3.2.3. Under the conditions of Theorem 3.2, if , then

[picture] | (3.17)

and

[picture] | (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.

References

[1]   | R.-F. Bai, F. Qi, and B.-Y. Xi, Hermite-Hadamard type inequalities for the m- and -logarithmically convex functions, Filomat 27(2013), no. 1, 1-7.
[2]   | P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequality point of view, Handbook of Analytic-computational Methods in Applied Mathematics, 135-200, Chapman & Hall/CRC, Boca Raton, FL, 2000.
[3]   | S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000; Available online at http://rgmia.org/monographs/hermite_hadamard.html.
[4]   | C. P. Niculescu and L.-E. Persson, Convex Functions and their Applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23. Springer, New York, 2006.
[5]   | F. Qi, Z.-L. Wei, and Q. Yang, Generalizations and refinements of Hermite-Hadamard's inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235-251.
[6]   | N. Ujević, Some double integral inequalities and applications, Acta Math. Univ. Comenian. (N.S.) 71 (2002), no. 2, 189-199.
[7]   | B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applica-tions, Hacet. J. Math. Stat. 42 (2013), no. 3, 243-257.