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Some Integral Inequalities of Hermite-Hadamard Type for ε-convex Functions

Jun Zhang, Zhi-Li Pei, Feng Qi
Turkish Journal of Analysis and Number Theory. 2017, 5(3), 117-120. DOI: 10.12691/tjant-5-3-5
Received January 07, 2017; Revised May 04, 2017; Accepted May 19, 2017

Abstract

In the paper, the authors establish a new integral identity. By this integral identity and Hölder’s inequality, the authors obtain some new inequalities of the Hermite-Hadamard type for ε-convex functions.

1. Introduction

We first list some definitions concerning various convex functions.

Definition 1.1. A functionis said to be convex if

(1.1)

holds for alland.

In 1 the concept of ε-convex functions was introduced as follows.

Definition 1.2. 1 A function and , if

(1.2)

is valid for all and, then we say that is a ε-convex function on .

The following inequalities of Hermite-Hadamard type were established for some of the above convex functions.

Theorem 1.1. 2 Let be differentiable on , with .

(i) If is convex function on , then

(1.3)

(ii) If is convex function on , , then

(1.4)

Theorem 1.2. 3 Let be differentiable on , with and .

(i) If is convex function on , then

(1.5)

(ii) If is convex function on , , then

(1.6)

and

(1.7)

In this paper, we establish a new integral identity. By this identity and Hölder’s inequality, some new Hermite-Hadamard type for the product of ε-convex function and discussed, some results are obtained.

2. Some Lemmas

Lemma 2.1 Let be differentiable onand where with , If , then the following identity holds:

(2.1)

Proof.

(2.2)

Since

(2.3)

Changing variables with

for , we get

(2.4)

Put the equalities (2.3) to (2.4) into the equality (2.2), the inequality (2.1) is thus proved. This completes of the proof.

By taking n= Lemma 2.1, we have the following identities.

Lemma 2.2 2 Let be differentiable onand where with , if , then

3. Some Integral Inequalities of Hermite-Hadamard Type

Now we are in a position to establish some new integral inequalities of Hermite-Hadamard type involving the ε-convex functions.

Theorem 3.1 Let be differentiable function on and , . If,then

(3.1)

Proof. Since is ε-convex function on , using the Lemma 2.1 and by the Hölder’s inequality, we have

Theorem 3.1 is proved.

Crollary3.1. Under the conditions of Theorem 3.1, then

(1) if , we have

(2) if , we have

Theorem 3.2 Let be differentiable function on and , and . If,then

(3.2)

Proof. Since is ε-convex function on , using the Lemma 2.1 and by the Hölder’s inequality, we have

(3.3)

Theorem 3.2 is proved.

Acknowledgements

This work was supported by the National Natural Science Foundation under Grant No. 61373067 and No. 61672301 of China and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.

References

[1]  D. H. Hyers, S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc., 3 (1952), 821-828.
In article      View Article
 
[2]  S. S. Dragomir, and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(1998), 91-95.
In article      View Article
 
[3]  C. E. M. Pearce and J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formula. Appl. Math. Lett., 13(2000), 51–55.
In article      View Article
 
[4]  F. Qi and B.-Y. Xi, Some Hermite–Hadamard type inequalities for geometrically quasi-convex functions, Proceedings of the Indian Academy of Sciences (Mathematical Sciences), 124:3(2014), 333-342.
In article      View Article
 
[5]  F. Qi, T.-Y. Zhang, and B.-Y. Xi, Integral inequalities of Hermite-Hadamard type for functions whose first derivatives are of convexity, Ukrainian Mathematical Journal, 66:5(2015), 625-640.
In article      View Article
 
[6]  B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions, Aequationes Math., 184:3(2012), 261-269.
In article      View Article
 
[7]  B.-Y. Xi and F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Scientiarum Mathematicarum Hungarica, 51:4(2014), 530-546;.
In article      View Article
 
[8]  B.-Y. Xi and F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal,. 16(2015), 873-890.
In article      View Article
 
[9]  B.-Y. Xi, S.-H. Wang, and F. Qi, Some inequalities for (h,m)-convex functions, Journal of Inequalities and Applications, 2014, 100, 12~pages.
In article      View Article
 
[10]  B.-Y. Xi, T.-Y. Zhang, and F. Qi, Some inequalities of Hermite--Hadamard type for m-harmonic-arithmetically convex functions, ScienceAsia, 41: 5(2015), 357-361.
In article      View Article
 

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Cite this article:

Normal Style
Jun Zhang, Zhi-Li Pei, Feng Qi. Some Integral Inequalities of Hermite-Hadamard Type for ε-convex Functions. Turkish Journal of Analysis and Number Theory. Vol. 5, No. 3, 2017, pp 117-120. http://pubs.sciepub.com/tjant/5/3/5
MLA Style
Zhang, Jun, Zhi-Li Pei, and Feng Qi. "Some Integral Inequalities of Hermite-Hadamard Type for ε-convex Functions." Turkish Journal of Analysis and Number Theory 5.3 (2017): 117-120.
APA Style
Zhang, J. , Pei, Z. , & Qi, F. (2017). Some Integral Inequalities of Hermite-Hadamard Type for ε-convex Functions. Turkish Journal of Analysis and Number Theory, 5(3), 117-120.
Chicago Style
Zhang, Jun, Zhi-Li Pei, and Feng Qi. "Some Integral Inequalities of Hermite-Hadamard Type for ε-convex Functions." Turkish Journal of Analysis and Number Theory 5, no. 3 (2017): 117-120.
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[1]  D. H. Hyers, S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc., 3 (1952), 821-828.
In article      View Article
 
[2]  S. S. Dragomir, and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(1998), 91-95.
In article      View Article
 
[3]  C. E. M. Pearce and J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formula. Appl. Math. Lett., 13(2000), 51–55.
In article      View Article
 
[4]  F. Qi and B.-Y. Xi, Some Hermite–Hadamard type inequalities for geometrically quasi-convex functions, Proceedings of the Indian Academy of Sciences (Mathematical Sciences), 124:3(2014), 333-342.
In article      View Article
 
[5]  F. Qi, T.-Y. Zhang, and B.-Y. Xi, Integral inequalities of Hermite-Hadamard type for functions whose first derivatives are of convexity, Ukrainian Mathematical Journal, 66:5(2015), 625-640.
In article      View Article
 
[6]  B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m- and (α,m)-geometrically convex functions, Aequationes Math., 184:3(2012), 261-269.
In article      View Article
 
[7]  B.-Y. Xi and F. Qi, Hermite-Hadamard type inequalities for geometrically r-convex functions, Studia Scientiarum Mathematicarum Hungarica, 51:4(2014), 530-546;.
In article      View Article
 
[8]  B.-Y. Xi and F. Qi, Inequalities of Hermite-Hadamard type for extended s-convex functions and applications to means, J. Nonlinear Convex Anal,. 16(2015), 873-890.
In article      View Article
 
[9]  B.-Y. Xi, S.-H. Wang, and F. Qi, Some inequalities for (h,m)-convex functions, Journal of Inequalities and Applications, 2014, 100, 12~pages.
In article      View Article
 
[10]  B.-Y. Xi, T.-Y. Zhang, and F. Qi, Some inequalities of Hermite--Hadamard type for m-harmonic-arithmetically convex functions, ScienceAsia, 41: 5(2015), 357-361.
In article      View Article