Research Article | Open Access

Feixiang Chen, Shanhe Wu, "Some Hermite-Hadamard Type Inequalities for Harmonically *s*-Convex Functions", *The Scientific World Journal*, vol. 2014, Article ID 279158, 7 pages, 2014. https://doi.org/10.1155/2014/279158

# Some Hermite-Hadamard Type Inequalities for Harmonically *s*-Convex Functions

**Academic Editor:**Wangmeng Zuo

#### Abstract

We establish some estimates of the right-hand side of Hermite-Hadamard type inequalities for functions whose derivatives absolute values are harmonically *s*-convex. Several Hermite-Hadamard type inequalities for products of two harmonically *s*-convex functions are also considered.

#### 1. Introduction

Let be a convex function and with ; then Inequality (1) is known as the Hermite-Hadamard inequality.

In [1], Hudzik and Maligranda considered the class of functions which are -convex in the second sense. This class of functions is defined as follows.

A function is said to be -convex in the second sense if the inequality holds for all , and for some fixed .

It can be easily seen that, for , -convexity reduces to ordinary convexity of functions defined on .

In [2], Dragomir and Fitzpatrick established a variant of Hermite-Hadamard inequality which holds for the -convex functions in the second sense.

Theorem 1 (see [2]). *Suppose that is an -convex function in the second sense, where and let , . If , then the following inequalities hold:
*

Some generalizations, improvements, and extensions of inequalities (1) and (3) can be found in the recent papers [2–18].

In [16], İşcan investigated the Hermite-Hadamard type inequalities for harmonically convex functions.

*Definition 2 (see [16]). *Let be a real interval. A function is said to be harmonically convex, if
for all and . If the inequality in (4) is reversed, then is said to be harmonically concave.

Theorem 3 (see [16]). *Let be a harmonically convex function and with . If , then one has
*

Theorem 4 (see [16]). *Let be a differentiable function on ( is the interior of ), with , and ; then
*

In [19], İşcan investigated the Hermite-Hadamard type inequalities for harmonically -convex functions.

*Definition 5 (see [19]). *Let be a real interval. A function is said to be harmonically -convex, if
for all , and for some fixed . If the inequality in (7) is reversed, then is said to be harmonically -concave.

Theorem 6 (see [19]). *Let be a harmonically -convex function and with . If , then one has
*

In [20], Pachpatte established two new Hermite-Hadamard type inequalities for products of convex functions asserted by Theorem 7.

Theorem 7 (see [20]). *Let and be real-valued, nonnegative, and convex functions on . Then
**
where and .*

For more results concerning the Hermite-Hadamard inequality, we refer the reader to [21–25] and the references cited therein.

In this paper, we establish some estimates of the right-hand side of Hermite-Hadamard type inequalities for functions whose derivatives absolute values are harmonically -convex. Moreover, we provide several Hermite-Hadamard type inequalities for products of two harmonically -convex functions.

#### 2. Inequalities for Harmonically -Convex Functions

We recall the following special functions.

The gamma function is as follows: the beta function is as follows: the hypergeometric function is as follows:

Our main results are given in the following theorems.

Theorem 8. *Let be a differentiable function on such that , where with . If is harmonically -convex on for some fixed , , then
**
where
*

*Proof. *Let . Using Theorem 4, the power mean inequality, and the harmonically -convexity of , we have
where
Calculating , , and , we find
Similarly, we get
This completes the proof of Theorem 8.

Theorem 9. *Let be a differentiable function on such that , where with . If is harmonically -convex on for some fixed , , then
**
where .*

*Proof. *Let . Utilizing Theorem 4, the Hölder inequality, and the harmonically -convexity of , we have
where
The proof of Theorem 9 is completed.

#### 3. Inequalities for Products of Harmonically -Convex Functions

Theorem 10. *Let , , , be functions such that . If is harmonically -convex and is harmonically -convex on for some fixed , then
**
where and .*

*Proof. *Since is harmonically -convex and is harmonically -convex on , then for we get
From (24), we get
Integrating both sides of the above inequality with respect to over , we obtain
The proof of Theorem 10 is completed.

*Remark 11. *Taking in Theorem 10, we obtain

*Remark 12. *Choosing and in Theorem 10 gives
which is the right-hand side inequality of (5).

Theorem 13. *Let , , , be functions such that . If is harmonically -convex and is harmonically -convex on for some fixed , then
**
where and .*

*Proof. *Using the harmonically -convexity of and , we have for all
Choosing and , we have
Integrating the resulting inequality with respect to over , we get
That is,
From
we get
This completes the proof of Theorem 13.

*Remark 14. *Putting in Theorem 13 gives

*Remark 15. *If we take and in Theorem 13, then we obtain

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The present investigation was supported, in part, by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11) and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049).

#### References

- H. Hudzik and L. Maligranda, “Some remarks on $s$-convex functions,”
*Aequationes Mathematicae*, vol. 48, no. 1, pp. 100–111, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. S. Dragomir and S. Fitzpatrick, “The Hadamard inequalities for $s$-convex functions in the second sense,”
*Demonstratio Mathematica*, vol. 32, no. 4, pp. 687–696, 1999. View at: Google Scholar | MathSciNet - S. Abramovich, G. Farid, and J. Pečarić, “More about Hermite-Hadamard inequalities, Cauchy's means, and superquadracity,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 102467, 14 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet - M. W. Alomari, M. Darus, and U. S. Kirmaci, “Some inequalities of Hermite-Hadamard type for $s$-convex functions,”
*Acta Mathematica Scientia B: English Edition*, vol. 31, no. 4, pp. 1643–1652, 2011. View at: Publisher Site | Google Scholar | MathSciNet - M. Avci, H. Kavurmaci, and M. E. Özdemir, “New inequalities of Hermite-Hadamard type via $s$-convex functions in the second sense with applications,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5171–5176, 2011. View at: Publisher Site | Google Scholar | MathSciNet - M. Bessenyei and Z. Páles, “Hadamard-type inequalities for generalized convex functions,”
*Mathematical Inequalities & Applications*, vol. 6, no. 3, pp. 379–392, 2003. View at: Publisher Site | Google Scholar | MathSciNet - J. Caballero and K. Sadarangani, “Hermite-Hadamard inequality for fuzzy integrals,”
*Applied Mathematics and Computation*, vol. 215, no. 6, pp. 2134–2138, 2009. View at: Publisher Site | Google Scholar | MathSciNet - F. Chen and X. Liu, “Refinements on the Hermite-Hadamard inequalities for $r$-convex functions,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 978493, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - F. Chen, “A note on Hermite-Hadamard inequalities for products of convex functions,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 935020, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - F. X. Chen, “On Hermite-Hadamard type inequalities for
*s*-convex functions on the coordinates via Riemann-Liouville fractional integrals,”*Journal of Applied Mathematics*, vol. 2014, Article ID 248710, 8 pages, 2014. View at: Publisher Site | Google Scholar - S. S. Dragomir, “On the Hadamard's inequlality for convex functions on the co-ordinates in a rectangle from the plane,”
*Taiwanese Journal of Mathematics*, vol. 5, no. 4, pp. 775–788, 2001. View at: Google Scholar - S. S. Dragomir, “Hermite-Hadamard's type inequalities for operator convex functions,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 766–772, 2011. View at: Publisher Site | Google Scholar - A. El Farissi, “Simple proof and refinement of Hermite-HADamard inequality,”
*Journal of Mathematical Inequalities*, vol. 4, no. 3, pp. 365–369, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Gao, “A note on the Hermite-Hadamard inequality,”
*Journal of Mathematical Inequalities*, vol. 4, no. 4, pp. 587–591, 2010. View at: Publisher Site | Google Scholar | MathSciNet - İ. İşcan and S. Wu, “Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals,”
*Applied Mathematics and Computation*, vol. 238, pp. 237–244, 2014. View at: Publisher Site | Google Scholar | MathSciNet - İ. İşcan, “Hermite-Hadamard and simpson-like type inequalities for differentiable harmonically convex functions,”
*Journal of Mathematics*, vol. 2014, Article ID 346305, 10 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - B. Sroysang, “Generalizations on some Hermite-Hadamard type inequalities for differentiable convex functions with applications to weighted means,”
*The Scientific World Journal*, vol. 2014, Article ID 717164, 13 pages, 2014. View at: Publisher Site | Google Scholar - B. Sroysang, “On the Hermite-Hadamard inequality and other integral inequalities involving several functions,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 921828, 6 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - İ. İşcan, “Ostrowski type inequalities for harmonically s-convex functions,” http://arxiv.org/abs/1307.5201. View at: Google Scholar
- B. G. Pachpatte, “On some inequalities for convex functions,”
*RGMIA Research Report Collection E*, vol. 6, 2003. View at: Google Scholar - B. G. Pachpatte, “A note on integral inequalities involving two log-convex functions,”
*Mathematical Inequalities & Applications*, vol. 4, no. 7, pp. 511–515, 2004. View at: Publisher Site | Google Scholar | MathSciNet - M. Z. Sarikaya, A. Saglam, and H. Yildirim, “On some Hadamard-type inequalities for $h$-convex functions,”
*Journal of Mathematical Inequalities*, vol. 2, no. 3, pp. 335–341, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Wu, “On the weighted generalization of the Hermite-HADamard inequality and its applications,”
*The Rocky Mountain Journal of Mathematics*, vol. 39, no. 5, pp. 1741–1749, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z. Xiao, Z. Zhang, and Y. Wu, “On weighted Hermite-Hadamard inequalities,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1147–1152, 2011. View at: Publisher Site | Google Scholar | MathSciNet - T. Zhang, A. Ji, and F. Qi, “On integral inequalities of Hermite-Hadamard type for $s$-geometrically convex functions,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 560586, 14 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Feixiang Chen and Shanhe Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.