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Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions

Chun-Long Li , Gui-Hua Gu, Bai-Ni Guo
Turkish Journal of Analysis and Number Theory. 2017, 5(6), 226-229. DOI: 10.12691/tjant-5-6-4
Received October 31, 2017; Revised December 01, 2017; Accepted December 06, 2017

Abstract

In the paper, by Holder’s integral inequality, the authors establish some Hermite-Hadamard type integral inequalities for harmonically quasi-convex functions.

1. Introduction

The following definitions for various convex functions are well known in the literature.

Definition 1.1 A function is said to be convex if

holds for all and .

Definition 1.2 ( 1, 2, 3) A function is said to be quasi-convex if

holds for all and .

Definition 1.3 ( 4) For with and if

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

Definition 1.4 ( 5) Let with and . If

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

Definition 1.5 ( 9) A function is said to be a harmonically quasi-convex function on if

holds for all and .

In 7, 8, the following inequalities of Hermite-Hadamard type were established.

Theorem 1.1 ( 7, Theorems 2.2 and 2.3]). Let be a differentiable mapping and with . Then

(i) if is convex on , then

(ii) if is convex on for , then

Theorem 1.2 ( 8, Theorem 2.3]) Let be differentiable on and with . If is s-convex on for , then

In this paper, we will create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.

2. A Lemma

For creating some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions, we need the following lemma.

Lemma 2.1 Let be differentiable on with , and . If , then

where

Proof Integrating by part and changing variables for yield

Lemma 2.1 is thus proved.

3. Some New Integral Inequalities of Hermite-Hadamard Type

Now we set off to create some integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions.

Theorem 3.1 Let be a differentiable function, with and . If is harmonically quasi-convex on , then

Proof Using Lemma 2.1 and the harmonic quasi-convexity of , we have

The proof of Theorem 3.1 is complete.

Theorem 3.2 Let be a differentiable function, with , and . If is harmonically quasi-convex on and , then

Proof Since is harmonically quasi-convex on , by Lemma 2.1 and Hölder's inequality, we have

Theorem 3.2 is thus proved.

Theorem 3.3. Let be a differentiable function, with , and . If is harmonically quasi-convex on for , then

Proof From the GA-inequality, we have

for all . By Lemma 2.1 and the harmonic quasi-convexity of and Hölder's inequality, we have

The proof of Theorem 3.3 is complete.

References

[1]  W. Fenchel, Convex cones, sets, and functions, Mimeographed Lectures Notes, Princeton University, Princeton, New Jersey, 1951.
In article      PubMed
 
[2]  K. L. Arrow and C. Enthovena, Quasi-concave programming, Econometrica, 1961, 29: 779-800.
In article      View Article
 
[3]  S. S. Dragomir, J. Pečaric and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335-341.
In article      View Article
 
[4]  G. Toader. Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Univ.Cluj-Napoca, Cluj-Napoca, 1985.
In article      PubMed
 
[5]  V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania), 1993.
In article      
 
[6]  Bo-Yan Xi, Tian-Yu Zhang, and Feng Qi. Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions. ScienceAsia, 2015, 41 (5): 357-361.
In article      View Article
 
[7]  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., 1998, 11: 91-95.
In article      View Article
 
[8]  U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146.
In article      View Article
 
[9]  Tian-Yu Zhang, Ai-Ping Ji, and Feng Qi. Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proceedings of the Jangjeon Mathematical Society, 2013, 16 (3), 399-407.
In article      View Article
 
[10]  S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55.
In article      View Article
 
[11]  S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babȩs-Bolyai Math. 38 (1993) 21-28.
In article      
 
[12]  Bo-Yan Xi and Feng Qi. Some new integral inequalities of Hermite-Hadamard type for (log, (α,m))-convex functions on co-ordinates. Studia Universitatis Babȩs-Bolyai Mathematica, 2015, 60 (4): 509-525.
In article      View Article
 
[13]  Bo-Yan Xi and Feng Qi. Integral inequalities of Hermite-Hadamard type for ((α,m), log)-convex functions on co-ordinates. Problemy Analiza-Issues of Analysis, 2015, 22 (2): 73-92.
In article      View Article
 
[14]  Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for geometrically r-convex functions. Studia Scientiarum Mathematicarum Hungarica, 2014, 51(4): 530-546.
In article      View Article
 
[15]  Bo-Yan Xi and Feng Qi. Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacettepe Journal of Mathematics and Statistics, 2013, 42(3): 243-257.
In article      View Article
 
[16]  Bo-Yan Xi and Feng Qi. Integral inequalities of Simpson type for logarithmically convex functions. Advanced Studies in Contemporary Mathematics, 2013, 23(4): 559-566.
In article      View Article
 
[17]  Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Functional Analysis and Applications, 2013, 18(2),: 163-176.
In article      View Article
 

Published with license by Science and Education Publishing, Copyright © 2017 Chun-Long Li, Gui-Hua Gu and Bai-Ni Guo

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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Normal Style
Chun-Long Li, Gui-Hua Gu, Bai-Ni Guo. Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions. Turkish Journal of Analysis and Number Theory. Vol. 5, No. 6, 2017, pp 226-229. http://pubs.sciepub.com/tjant/5/6/4
MLA Style
Li, Chun-Long, Gui-Hua Gu, and Bai-Ni Guo. "Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions." Turkish Journal of Analysis and Number Theory 5.6 (2017): 226-229.
APA Style
Li, C. , Gu, G. , & Guo, B. (2017). Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions. Turkish Journal of Analysis and Number Theory, 5(6), 226-229.
Chicago Style
Li, Chun-Long, Gui-Hua Gu, and Bai-Ni Guo. "Some Inequalities of the Hermite-Hadamard Type for Harmonically Quasi-Convex Functions." Turkish Journal of Analysis and Number Theory 5, no. 6 (2017): 226-229.
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[1]  W. Fenchel, Convex cones, sets, and functions, Mimeographed Lectures Notes, Princeton University, Princeton, New Jersey, 1951.
In article      PubMed
 
[2]  K. L. Arrow and C. Enthovena, Quasi-concave programming, Econometrica, 1961, 29: 779-800.
In article      View Article
 
[3]  S. S. Dragomir, J. Pečaric and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335-341.
In article      View Article
 
[4]  G. Toader. Some generalizations of the convexity. Proceedings of the Colloquium on Approximation and Optimization, Univ.Cluj-Napoca, Cluj-Napoca, 1985.
In article      PubMed
 
[5]  V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania), 1993.
In article      
 
[6]  Bo-Yan Xi, Tian-Yu Zhang, and Feng Qi. Some inequalities of Hermite-Hadamard type for m-harmonic-arithmetically convex functions. ScienceAsia, 2015, 41 (5): 357-361.
In article      View Article
 
[7]  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., 1998, 11: 91-95.
In article      View Article
 
[8]  U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146.
In article      View Article
 
[9]  Tian-Yu Zhang, Ai-Ping Ji, and Feng Qi. Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proceedings of the Jangjeon Mathematical Society, 2013, 16 (3), 399-407.
In article      View Article
 
[10]  S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55.
In article      View Article
 
[11]  S. S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babȩs-Bolyai Math. 38 (1993) 21-28.
In article      
 
[12]  Bo-Yan Xi and Feng Qi. Some new integral inequalities of Hermite-Hadamard type for (log, (α,m))-convex functions on co-ordinates. Studia Universitatis Babȩs-Bolyai Mathematica, 2015, 60 (4): 509-525.
In article      View Article
 
[13]  Bo-Yan Xi and Feng Qi. Integral inequalities of Hermite-Hadamard type for ((α,m), log)-convex functions on co-ordinates. Problemy Analiza-Issues of Analysis, 2015, 22 (2): 73-92.
In article      View Article
 
[14]  Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for geometrically r-convex functions. Studia Scientiarum Mathematicarum Hungarica, 2014, 51(4): 530-546.
In article      View Article
 
[15]  Bo-Yan Xi and Feng Qi. Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacettepe Journal of Mathematics and Statistics, 2013, 42(3): 243-257.
In article      View Article
 
[16]  Bo-Yan Xi and Feng Qi. Integral inequalities of Simpson type for logarithmically convex functions. Advanced Studies in Contemporary Mathematics, 2013, 23(4): 559-566.
In article      View Article
 
[17]  Bo-Yan Xi and Feng Qi. Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Functional Analysis and Applications, 2013, 18(2),: 163-176.
In article      View Article