- Research Article
- Open Access

# Multiplicative Concavity of the Integral of Multiplicatively Concave Functions

- Yu-Ming Chu
^{1}Email author and - Xiao-Ming Zhang
^{2}

**2010**:845390

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

© Y.-M. Chu and X.-M. Zhang. 2010

**Received:**25 March 2010**Accepted:**7 June 2010**Published:**28 June 2010

## Abstract

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

## Keywords

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

## 1. Introduction

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

Definition 1.1.

for all . And is called concave if is convex.

Definition 1.2.

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

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.

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

Definition 1.5.

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.

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]).

and ,

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

Lemma 2.3.

Lemma 2.4 (see [2]).

for all .

Lemma 2.5.

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

Proof.

for all .

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

Case 1.

Case 2.

for all .

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

Case 3.

Lemma 2.6.

Proof.

We divide the proof into five cases.

Case 1.

Case 2.

Case 3.

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

Case 4.

Case 5.

Lemma 2.7.

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

Proof.

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 .

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

## Declarations

### 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.

## Authors’ Affiliations

## References

- Montel P: Sur les functions convexes et les fonctions sousharmoniques.
*Journal de Mathématiques*1928, 7(9):29–60.MATHGoogle Scholar - Niculescu CP: Convexity according to the geometric mean.
*Mathematical Inequalities & Applications*2000, 3(2):155–167.MathSciNetView ArticleMATHGoogle Scholar - Niculescu CP: Convexity according to means.
*Mathematical Inequalities & Applications*2003, 6(4):551–579.MathSciNetMATHGoogle Scholar - 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.View ArticleGoogle Scholar - 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.2007.02.016MathSciNetView ArticleMATHGoogle Scholar - Finol CE, Wójtowicz M: Multiplicative properties of real functions with applications to classical functions.
*Aequationes Mathematicae*2000, 59(1–2):134–149.MathSciNetView ArticleMATHGoogle Scholar - 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.Google Scholar - Guan KZ: A class of symmetric functions for multiplicatively convex function.
*Mathematical Inequalities & Applications*2007, 10(4):745–753.MathSciNetView ArticleMATHGoogle Scholar - 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.2007.197MathSciNetView ArticleMATHGoogle Scholar - Chu Y, Zhang XM, Wang G-D: The Schur geometrical convexity of the extended mean values.
*Journal of Convex Analysis*2008, 15(4):707–718.MathSciNetMATHGoogle Scholar - Chu YM, Xia WF, Zhao TH: Schur convexity for a class of symmetric functions.
*Science China Mathematics*2010, 53(2):465–474. 10.1007/s11425-009-0188-2MathSciNetView ArticleMATHGoogle Scholar - Xia W-F, Chu Y-M: Schur-convexity for a class of symmetric functions and its applications.
*Journal of Inequalities and Applications*2009, 2009:-15.Google Scholar - Zhang X-M, Chu Y-M: A double inequality for gamma function.
*Journal of Inequalities and Applications*2009, 2009:-7.Google Scholar - 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.Google Scholar - 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 - 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.2008.11.068MathSciNetView ArticleMATHGoogle Scholar - Zhang 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.MathSciNetMATHGoogle Scholar - Zhang X-M, Chu Y-M: A new method to study analytic inequalities.
*Journal of Inequalities and Applications*2010, 2010:-13.Google Scholar - 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.Google Scholar - Koliha JJ: Approximation of convex functions.
*Real Analysis Exchange*2003–2004, 29(1):465–471.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.