Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

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

Jun Zhang, Zhi-Li Pei, Feng Qi*

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China,

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.

Keywords: Hermite-Hadamard type type inequality, Integral identity, ε-convex function

Received January 07, 2017; Revised May 04, 2017; Accepted May 19, 2017

Citation: 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, 5 (3). 117-120. January 2017. doi:10.12691/tjant-5-3-5

1. Introduction

We first list some definitions concerning various convex functions.

Definition 1.1. A functionis said to be convex if

[picture] | (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

[picture] | (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

[picture] | (1.3)

(ii) If is convex function on , , then

[picture] | (1.4)

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

(i) If is convex function on , then

[picture] | (1.5)

(ii) If is convex function on , , then

[picture] | (1.6)

and

[picture] | (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:

[picture] | (2.1)

Proof.

[picture] | (2.2)

Since

[picture] | (2.3)

Changing variables with

for , we get

[picture] | (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

[picture] | (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

[picture] | (3.2)

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

[picture] | (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.
[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.
[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.
[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.
[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.
[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.
[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;.
[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.
[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.
[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.