Skip to main content

The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application

Abstract

We established a new Hermit-Hadamard type inequality for GA-convex functions. As applications, we obtain two new Gautschi type inequalities for gamma function.

1. Introduction

Let be a convex (concave) function on ; the well-known Hermite-Hadamard's inequality [1] can be expressed as

(1.1)

Recently, Hermite-Hadamard's inequality has been the subject of intensive research. In particular, many improvements, generalizations, and applications for the Hermite-Hadamard's inequality can be found in the literature [220].

Let be an interval; a real-valued function is said to be GA-convex (concave) on if for all and .

In [21], Anderson et al. discussed the GA and related kinds of convexity; some applications to special functions were presented.

For , let , , and be the geometric, logarithmic, identric, and arithmetic means of and , respectively. Then

(1.2)

The first purpose of this paper is to establish the following new Hermite-Hadamard type inequality for GA-convex (concave) functions.

Theorem 1.1.

If and is a differentiable GA-convex (concave) function, then

(1.3)

For real and positive values of , the Euler gamma function and its logarithmic derivative , the so-called digamma function, are defined by

(1.4)

The ratio has attracted the attention of many mathematicians and physicists. Gautschi [22] first proved that

(1.5)

for and

A strengthened upper bound was given by Erber [23]:

(1.6)

In [24], Kečkić and Vasić established the following double inequality for :

(1.7)

In [25], Kershaw obtained

(1.8)

for and .

In [26], Zhang and Chu proved

(1.9)

for all .

In [27], Zhang and Chu presented

(1.10)

for all .

The second purpose of this paper is to establish the following two new Gautschi type inequalities by using Theorem 1.1.

Theorem 1.2.

If , then

(1.11)

Theorem 1.3.

If , then

(1.12)

2. Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

Lemma 2.1.

One has

Proof.

Simple computations lead to

(2.1)

Lemma 2.2 (see [28, Lemma ]).

If , then

(2.2)
(2.3)

where , , , ,  ,  ,  .

Lemma 2.3.

Suppose that is an interval and is a real-valued function. If is second-order differentiable on , then is GA-convex (concave) on if and only if

(2.4)

for all .

Proof.

Lemma 2.3 follows easily from the basic properties of convex (concave) functions and the fact that is GA-convex (concave) on if and only if is convex (concave) on .

Lemma 2.4 (see [29, Theorem ]).

If , then

(2.5)

Lemma 2.5.

is GA-concave on .

Proof.

Differentiating the well-known identity we get

(2.6)

From inequalities (2.5) and (2.6) we have

(2.7)

Inequality (2.7) leads to

(2.8)

Therefore, Lemma 2.5 follows from (2.8) and Lemma 2.3.

Lemma 2.6.

is GA-convex on .

Proof.

Simple computation leads to

(2.9)

From (2.9) and Lemma 2.3 we know that we need only to prove that

(2.10)

We divide the proof into three cases.

Case 1.

. Taking in (2.2) and in (2.3) we get

(2.11)
(2.12)

Inequalities (2.11) and (2.12) together with lead to

(2.13)

Case 2.

. It is well-known that

(2.14)

where is Euler's constant.

Differentiating (2.14) we get

(2.15)
(2.16)

We clearly see that is increasing in for ; hence (2.15) and (2.16) lead to

(2.17)

It follows from inequality (2.17), Lemma 2.1, and that

(2.18)

Case 3.

. Since is decreasing in for , hence (2.15) and (2.16) imply that

(2.19)

From (2.19), Lemma 2.1, and we get

(2.20)

It is not difficult to verify that

(2.21)

Therefore, inequality (2.10) follows from (2.20) and (2.21).

3. Proof of Theorems 1.1, 1.2, and 1.3

Proof of Theorem 1.1.

Suppose that is a GA-convex function. For any fixed , if , then is convex on and

(3.1)

Inequality (3.1) implies that

(3.2)

Let , then inequality (3.2) leads to that for . Hence , namely,

(3.3)

Using a similar method we get

(3.4)

Let , then

(3.5)

From inequalities (3.3) and (3.4) together with (3.5) we clearly see that

(3.6)

Next for any , let , then and . From the definition of GA-convex function and the transformation to variable of integration we get

(3.7)

Therefore, Theorem 1.1 follows from inequalities (3.6) and (3.7).

Proof of Theorem 1.2.

From Lemmas 2.5 and 2.6 together with Theorem 1.1 we clearly see that

(3.8)
(3.9)

Therefore, Theorem 1.2 follows from (3.8) and (3.9).

Proof of Theorem 1.3.

From Lemmas 2.5 and 2.6 together with Theorem 1.1 we get

(3.10)
(3.11)

Inequalities (3.10) and (3.11) lead to

(3.12)
(3.13)

Therefore, Theorem 1.3 follows from (3.12) and (3.13).

Remark 3.1.

Making use of a computer and the mathematica software we can show that the bounds in Theorems 1.2 and 1.3 are stronger than that in inequalities (1.9) and (1.10) for some and . In fact, if we let , , , , , and , then we have Tables 1 and 2 via elementary computation.

Table 1 Comparison of and with and for some and
Table 2 Comparison of and with for some and .

Remark 3.2.

We clear see that the lower bound in Theorem 1.3 is stronger than that in inequality (1.9) for all .

References

  1. Hadamard J: Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées 1893, 58: 171–215.

    Google Scholar 

  2. Niculescu CP: The Hermite-Hadamard inequality for convex functions on global NPC space. Journal of Mathematical Analysis and Applications 2009, 356(1):295–301. 10.1016/j.jmaa.2009.03.007

    MATH  MathSciNet  Article  Google Scholar 

  3. Wu S-H: On the weighted generalization of the Hermite-Hadamard inequality and its applications. The Rocky Mountain Journal of Mathematics 2009, 39(5):1741–1749. 10.1216/RMJ-2009-39-5-1741

    MATH  MathSciNet  Article  Google Scholar 

  4. Alomari M, Darus M: On the Hadamard's inequality for log-convex functions on the coordinates. Journal of Inequalities and Applications 2009, 2009:-13.

    Google Scholar 

  5. Dinu C: Hermite-Hadamard inequality on time scales. Journal of Inequalities and Applications 2008, 2008:-24.

    Google Scholar 

  6. Bessenyei M: The Hermite-Hadamard inequality on simplices. American Mathematical Monthly 2008, 115(4):339–345.

    MATH  MathSciNet  Google Scholar 

  7. Mihăilescu M, Niculescu CP: An extension of the Hermite-Hadamard inequality through subharmonic functions. Glasgow Mathematical Journal 2007, 49(3):509–514.

    MATH  MathSciNet  Google Scholar 

  8. Bessenyei M, Páles Z: Characterization of convexity via Hadamard's inequality. Mathematical Inequalities & Applications 2006, 9(1):53–62.

    MATH  MathSciNet  Article  Google Scholar 

  9. Yang G-S, Hwang D-Y, Tseng K-L: Some inequalities for differentiable convex and concave mappings. Computers & Mathematics with Applications 2004, 47(2–3):207–216. 10.1016/S0898-1221(04)90017-X

    MATH  MathSciNet  Article  Google Scholar 

  10. Sun M, Yang X: Generalized Hadamard's inequality and -convex functions in Carnot groups. Journal of Mathematical Analysis and Applications 2004, 294(2):387–398. 10.1016/j.jmaa.2003.10.050

    MATH  MathSciNet  Article  Google Scholar 

  11. Wang L: On extensions and refinements of Hermite-Hadamard inequalities for convex functions. Mathematical Inequalities & Applications 2003, 6(4):659–666.

    MATH  MathSciNet  Article  Google Scholar 

  12. Mercer AM: Hadamard's inequality for a triangle, a regular polygon and a circle. Mathematical Inequalities & Applications 2002, 5(2):219–223.

    MATH  MathSciNet  Article  Google Scholar 

  13. Dragomir SS, Pearce CEM: Quasilinearity & Hadamard's inequality. Mathematical Inequalities & Applications 2002, 5(3):463–471.

    MATH  MathSciNet  Article  Google Scholar 

  14. Dragomir SS, Cho YJ, Kim SS: Inequalities of Hadamard's type for Lipschitzian mappings and their applications. Journal of Mathematical Analysis and Applications 2000, 245(2):489–501. 10.1006/jmaa.2000.6769

    MATH  MathSciNet  Article  Google Scholar 

  15. Yang G-S, Tseng K-L: On certain integral inequalities related to Hermite-Hadamard inequalities. Journal of Mathematical Analysis and Applications 1999, 239(1):180–187. 10.1006/jmaa.1999.6506

    MATH  MathSciNet  Article  Google Scholar 

  16. Dragomir SS, Agarwal RP: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied Mathematics Letters 1998, 11(5):91–95. 10.1016/S0893-9659(98)00086-X

    MATH  MathSciNet  Article  Google Scholar 

  17. Dragomir SS, Agarwal RP: Two new mappings associated with Hadamard's inequalities for convex functions. Applied Mathematics Letters 1998, 11(3):33–38. 10.1016/S0893-9659(98)00030-5

    MATH  MathSciNet  Article  Google Scholar 

  18. Pearce CEM, Pečarić J, Šimić V: Stolarsky means and Hadamard's inequality. Journal of Mathematical Analysis and Applications 1998, 220(1):99–109. 10.1006/jmaa.1997.5822

    MATH  MathSciNet  Article  Google Scholar 

  19. Gill PM, Pearce CEM, Pečarić J: Hadamard's inequality for -convex functions. Journal of Mathematical Analysis and Applications 1997, 215(2):461–470. 10.1006/jmaa.1997.5645

    MATH  MathSciNet  Article  Google Scholar 

  20. Pečarić JE, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.

    Google Scholar 

  21. Anderson GD, Vamanamurthy MK, Vuorinen M: Generalized convexity and inequalities. Journal of Mathematical Analysis and Applications 2007, 335(2):1294–1308. 10.1016/j.jmaa.2007.02.016

    MATH  MathSciNet  Article  Google Scholar 

  22. Gautschi W: Some elementary inequalities relating to the gamma and incomplete gamma function. Journal of Mathematics and Physics 1959, 38: 77–81.

    MATH  MathSciNet  Article  Google Scholar 

  23. Erber T: The gamma function inequalities of Gurland and Gautschi. Skandinavisk Aktuarietidskrift 1961, 1960: 27–28.

    MATH  MathSciNet  Google Scholar 

  24. Kečkić JD, Vasić PM: Some inequalities for the gamma function. Publications de l'Institut Mathématique 1971, 11(25):107–114.

    Google Scholar 

  25. Kershaw D: Some extensions of W. Gautschi's inequalities for the gamma function. Mathematics of Computation 1983, 41(164):607–611.

    MATH  MathSciNet  Google Scholar 

  26. Zhang X, Chu Y: An inequality involving the gamma function and the psi function. International Journal of Modern Mathematics 2008, 3(1):67–73.

    MATH  MathSciNet  Google Scholar 

  27. Zhang X, Chu Y: A double inequality for gamma function. Journal of Inequalities and Applications 2009, 2009:-7.

    Google Scholar 

  28. Zhao T-H, Chu Y-M, Jiang Y-P: Monotonic and logarithmically convex properties of a function involving gamma functions. Journal of Inequalities and Applications 2009, 2009:-13.

    Google Scholar 

  29. Elbert Á, Laforgia A: On some properties of the gamma function. Proceedings of the American Mathematical Society 2000, 128(9):2667–2673. 10.1090/S0002-9939-00-05520-9

    MATH  MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

The authors wish to thank the anonymous referee for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu-Ming Chu.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhang, XM., Chu, YM. & Zhang, XH. The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application. J Inequal Appl 2010, 507560 (2010). https://doi.org/10.1155/2010/507560

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2010/507560

Keywords

  • Special Function
  • Simple Computation
  • Gamma Function
  • Elementary Computation
  • Intensive Research