Multiplicative Concavity of the Integral of Multiplicatively Concave Functions

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

(1.1)

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

(1.2)

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

For and , we denote

(1.3)

(1.4)

For , , we denote

(1.5)

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

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

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

(2.1)

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

(2.2)

, 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

(2.3)

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

(2.4)

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

(2.5)

for all .

Next, set

(2.6)

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

(2.7)

Then from the multiplicative concavity of we clearly see that

(2.8)

for all .

From (2.7) and (2.8) we get

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

(2.13)

(2.14)

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

(2.15)

Combining (2.12) and (2.15) we get

(2.16)

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

(2.17)

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

(2.18)

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