- Research Article
- Open Access
- Published:

# Multiplicative Concavity of the Integral of Multiplicatively Concave Functions

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 845390 (2010)

## Abstract

We prove that is multiplicatively concave on if is continuous and multiplicatively concave.

## 1. Introduction

For convenience of the readers, we first recall some definitions and notations as follows.

Definition 1.1.

Let be an interval. A real-valued function is said to be convex if

for all . And is called concave if is convex.

Definition 1.2.

Let be an interval. A real-valued function is said to be multiplicatively convex if

for all . And is called multiplicatively concave if is multiplicatively convex.

For and , we denote

For , , we denote

Definition 1.3.

A set is said to be convex if whenever . And a set is said to be multiplicatively convex if whenever .

From Definition 1.3 we clearly see that is a multiplicatively convex set if and only if is a convex set, and is a convex set if and only if is a multiplicatively convex set.

Definition 1.4.

Let be a convex set. A real-valued function is said to be convex if

for all . And is said to be concave if is convex.

Definition 1.5.

Let be a multiplicatively convex set. A real-valued function is said to be multiplicatively convex if

for all . And is called multiplicatively concave if is multiplicatively convex.

From Definitions 1.1 and 1.2, the following Theorem A is obvious.

Theorem A.

Suppose that is a subinterval of and is multiplicatively convex. Then

is convex. Conversely, if is an interval and is convex, then

is multiplicatively convex.

Equivalently, is a multiplicatively convex function if and only if is a convex function of . Modulo this characterization, the class of all multiplicatively convex functions was first considered by Motel [1], in a beautiful paper discussing the analogues of the notion of convex function in variables. However, the roots of the research in this area can be traced long before him. In a long time, the subject of multiplicative convexity seems to be even forgotten, which is a pity because of its richness. Recently, the multiplicative convexity has been the subject of intensive research. In particular, many remarkable inequalities were found via the approach of multiplicative convexity (see [2–18]).

The main purpose of this paper is to prove Theorem 1.6.

Theorem 1.6.

If is continuous and multiplicatively concave, then is multiplicatively concave on .

## 2. Lemmas and the Proof of Theorem 1.6

For the sake of readability, we first introduce and establish several lemmas which will be used to predigest the proof of Theorem 1.6.

Lemma 2.1 can be derived from Definitions 1.4 and 1.5.

Lemma 2.1.

If is a multiplicatively convex set, and is multiplicatively convex (or concave, resp.), then is convex (or concave, resp.) on . Conversely, if is a convex set, and is convex (or concave, resp.), then is multiplicatively convex (or concave, resp.) on .

Lemma 2.2 (see [19]).

If is a convex set, and is second-order differentiable, then is convex (or concave, resp.) if and only if is a positive (or negative, resp.) semidefinite matrix for all . Here

and ,

Making use of Lemmas 2.1 and 2.2 we get the following Lemma 2.3.

Lemma 2.3.

If is a multiplicatively convex set, and is second-order differentiable, then is multiplicatively convex (or concave, resp.) if and only if is a positive (or negative, resp.) semidefinite matrix for all . Here

, and

Lemma 2.4 (see [2]).

If is an interval and is differentiable, then is multiplicatively convex (or concave, resp.) if and only if is increasing (or decreasing, resp.) on . If moreover is second-order differentiable, then is multiplicaively convex (or concave, resp.) if and only if

for all .

Lemma 2.5.

Suppose that is a second-order differentiable multiplicatively concave function. If , then is also multiplicatively concave on .

Proof.

For , from the expression of we get

According to Lemma 2.4, to prove that is multiplicatively concave on , it is sufficient to prove that

for all .

Next, set

From Lemma 2.4 we know that is decreasing; the following three cases will complete the proof of inequality (2.5).

Case 1.

Then , and for all ; hence (2.5) is true for all .

Case 2.

. Then , that is, for all .

Let

Then from the multiplicative concavity of we clearly see that

for all .

From (2.7) and (2.8) we get

for all . Therefore, inequality (2.5) follows from (2.7) and (2.9).

Case 3.

and . Then there exists a unique such that and for . Making use of the similar argument as in Case 2 we know that inequality (2.5) holds for ; this result and imply that (2.5) holds for all .

Lemma 2.6.

If is a second-order differentiable multiplicatively concave function, then

Proof.

We divide the proof into five cases.

Case 1.

. Then from Lemma 2.4 we know that is decreasing on ; hence we get It is obvious that inequality (2.10) holds in this case.

Case 2.

Then (2.10) follows from

Case 3.

Then for all . From (2.7) and (2.8) we get

Therefore, inequality (2.10) follows from inequality (2.11) and

Case 4.

Then for all ; hence inequality (2.10) follows from (2.11) and

Case 5.

. Then we clearly see that (2.10) is true.

Lemma 2.7.

If is a second-order differentiable multiplicatively concave function, then is multiplicatively concave on .

Proof.

For , without loss of generality, we assume that . Then simple computations lead to

From Lemma 2.5 we know that is multiplicatively concave; then Lemma 2.4 leads to

Combining (2.12) and (2.15) we get

Equations (2.12)–(2.14) and Lemma 2.6 yield

Therefore, Lemma 2.7 follows from (2.16) and (2.17) together with Lemma 2.3.

Lemma 2.8 (see [20]).

For each continuous convex function , there exists a sequence of infinitely differentiable convex functions , such that converges uniformly to on .

From Definitions 1.1 and 1.2, Theorem A, and Lemma 2.8 we can get Lemma 2.9 immediately.

Lemma 2.9.

For each continuous multiplicatively convex (or concave, resp.) function , there exists a sequence of infinitely differentiable multiplicatively convex (or concave, resp.) functions , such that converges uniformly to on .

Proof of Theorem 1.6.

Since is a continuous multiplicatively concave function, from Lemma 2.9 we know that there exists a sequence of infinitely differentiable multiplicatively concave function , such that converges uniformly to on .

For , taking , then by Lemma 2.7 we clearly see that is multiplicatively concave on and

Therefore, Theorem 1.6 follows from Definition 1.5 and (2.18).

## References

Montel P: Sur les functions convexes et les fonctions sousharmoniques.

*Journal de Mathématiques*1928, 7(9):29–60.Niculescu CP: Convexity according to the geometric mean.

*Mathematical Inequalities & Applications*2000, 3(2):155–167.Niculescu CP: Convexity according to means.

*Mathematical Inequalities & Applications*2003, 6(4):551–579.Niculescu CP, Persson L-E:

*Convex Functions and Their Applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23*. Springer, New York, NY, USA; 2006:xvi+255.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.2007.02.016Finol CE, Wójtowicz M: Multiplicative properties of real functions with applications to classical functions.

*Aequationes Mathematicae*2000, 59(1–2):134–149.Trif T: Convexity of the gamma function with respect to Hölder means. In

*Inequality Theory and Applications. Vol. 3*. Nova Science Publishers, Hauppauge, NY, USA; 2003:189–195.Guan KZ: A class of symmetric functions for multiplicatively convex function.

*Mathematical Inequalities & Applications*2007, 10(4):745–753.Zhang X-M, Yang Z-H: Differential criterion of -dimensional geometrically convex functions.

*Journal of Applied Analysis*2007, 13(2):197–208. 10.1515/JAA.2007.197Chu Y, Zhang XM, Wang G-D: The Schur geometrical convexity of the extended mean values.

*Journal of Convex Analysis*2008, 15(4):707–718.Chu YM, Xia WF, Zhao TH: Schur convexity for a class of symmetric functions.

*Science China Mathematics*2010, 53(2):465–474. 10.1007/s11425-009-0188-2Xia W-F, Chu Y-M: Schur-convexity for a class of symmetric functions and its applications.

*Journal of Inequalities and Applications*2009, 2009:-15.Zhang X-M, Chu Y-M: A double inequality for gamma function.

*Journal of Inequalities and Applications*2009, 2009:-7.Chu Y-M, Lv Y-P: The Schur harmonic convexity of the Hamy symmetric function and its applications.

*Journal of Inequalities and Applications*2009, 2009:-10.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.Zhang X-H, Wang G-D, Chu Y-M: Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions.

*Journal of Mathematical Analysis and Applications*2009, 353(1):256–259. 10.1016/j.jmaa.2008.11.068Zhang X-M, Chu Y-M: The geometrical convexity and concavity of integral for convex and concave functions.

*International Journal of Modern Mathematics*2008, 3(3):345–350.Zhang X-M, Chu Y-M: A new method to study analytic inequalities.

*Journal of Inequalities and Applications*2010, 2010:-13.Marshall AW, Olkin I:

*Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering*.*Volume 143*. Academic Press, New York, NY, USA; 1979:xx+569.Koliha JJ: Approximation of convex functions.

*Real Analysis Exchange*2003–2004, 29(1):465–471.

## Acknowledgments

The research was supported by the Natural Science Foundation of China under Grant 60850005 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

## Author information

### Affiliations

### Corresponding author

## 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.

## About this article

### Cite this article

Chu, YM., Zhang, XM. Multiplicative Concavity of the Integral of Multiplicatively Concave Functions.
*J Inequal Appl* **2010, **845390 (2010). https://doi.org/10.1155/2010/845390

Received:

Accepted:

Published:

DOI: https://doi.org/10.1155/2010/845390

### Keywords

- Convex Function
- Simple Computation
- Intensive Research
- Concave Function
- Remarkable Inequality