On The Hadamard's Inequality for Log-Convex Functions on the Coordinates

Abstract

Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given.

1. Introduction

Let be a convex mapping defined on the interval of real numbers and , with , then

(1.1)

holds, this inequality is known as the Hermite-Hadamard inequality. For refinements, counterparts, generalizations and new Hadamard-type inequalities, see [1–8].

A positive function is called log-convex on a real interval , if for all and ,

(1.2)

If is a positive log-concave function, then the inequality is reversed. Equivalently, a function is log-convex on if is positive and is convex on . Also, if and exists on , then is log-convex if and only if .

The logarithmic mean of the positive real numbers , , is defined as

(1.3)

A version of Hadamard's inequality for log-convex (concave) functions was given in [9], as follows.

Theorem 1.1.

Suppose that is a positive log-convex function on , then

(1.4)

If is a positive log-concave function, then the inequality is reversed.

For refinements, counterparts and generalizations of log-convexity see [9–13].

A convex function on the coordinates was introduced by Dragomir in [8]. A function which is convex in is called coordinated convex on if the partial mapping , and , , are convex for all and .

An inequality of Hadamard's type for coordinated convex mapping on a rectangle from the plane established by Dragomir in [8], is as follows.

Theorem 1.2.

Suppose that is coordinated convex on , then

(1.5)

The above inequalities are sharp.

The maximum modulus principle in complex analysis states that if is a holomorphic function, then the modulus cannot exhibit a true local maximum that is properly within the domain of . Characterizations of the maximum principle for sub(super)harmonic functions are considered in [14], as follows.

Theorem 1.3.

Let be a region and let be a sub(super)harmonic function. If there is a point with , for all then is a constant function.

Theorem 1.4.

Let be a region and let and be bounded real-valued functions defined on such that is subharmonic and is superharmonic. If for each point

(1.6)

then for all or and is harmonic.

In this paper, a new version of the maximum (minimum) principle in terms of convexity, and some inequalities of the Hadamard type are obtained.

2. On Coordinated Convexity and Sub(Super)Harmonic Functions

Consider the -dimensional interval in . A function is called convex in if

(2.1)

holds for all and .

As in [8], we define a log-convex function on the coordinates as follows: a function will be called coordinated log-convex on if the partial mappings , and , , are log-convex for all and . A formal definition of a coordinated log-convex function may be stated as follows.

Definition 2.1.

A function will be called coordinated log-convex on , for all and , if the following inequality holds,

(2.2)

Equivalently, we can determine whether or not the function is coordinated log-convex by using the following lemma.

Lemma 2.2.

Let . If is twice differentiable then is coordinated log-convex on if and only if for the functions , defined by and , defined by , we have

(2.3)

Proof.

The proof is straight forward using the elementary properties of log-convexity in one variable.

Proposition 2.3.

Suppose that is twice differentiable on and log-convex on and is twice differentiable on and log-convex on . Let be a twice differentiable function defined by , then is coordinated log-convex on .

Proof.

This follows directly using Lemma 2.2.

The following result holds.

Proposition 2.4.

Every log-convex function is log-convex on the coordinates, but the converse is not generally true.

Proof.

Suppose that is convex in . Consider the function , , then for , and , we have

(2.4)

which shows the log-convexity of . The proof that , , is also log-convex on for all follows likewise. Now, consider the mapping given by . It is obvious that is log-convex on the coordinates but not log-convex on . Indeed, if and , we have:

(2.5)

Thus, for all and , we have

(2.6)

which shows that is not log-convex on .

In the following, a Jensen-type inequality for coordinated log-convex functions is considered.

Proposition 2.5.

Let be a positive coordinated log-convex function on the open set and let , . If and , , then

(2.7)

Proof.

Let , be such that , and let , be such that , then we have,

(2.8)

and, since is positive,

(2.9)

which is as required.

Remark 2.6.

Let , then the following inequality holds:

(2.10)

The above result may be generalized to the integral form as follows.

Proposition 2.7.

Let be a positive coordinated log-convex function on the and let be integrable with , and let be integrable with . If is positive, , and is integrable on and is positive, , and is integrable on then

(2.11)

Proof.

Applying Jensen's integral inequality in one variable on the -coordinate and on the -coordinate we get the required result. The details are omitted.

Theorem 2.8.

Let be a positive coordinated log-convex function in , then for all distinct , such that and distinct such that , the following inequality holds:

(2.12)

Proof.

Let be distinct points in and let be distinct points in . Setting , and let , , we have

(2.13)

and we can write

(2.14)

From this inequality it is easy to deduce the required result (2.12).

The subharmonic functions exhibit many properties of convex functions. Next, we give some results for the coordinated convexity and sub(super)harmonic functions.

Proposition 2.9.

Let be coordinated convex (concave) on . If is a twice differentiable on , then is sub(super)harmonic on .

Proof.

Since is coordinated convex on then the partial mappings , and , , are convex for all and . Equivalently, since is differentiable we can write

(2.15)

for all , and

(2.16)

for all , which imply that

(2.17)

which shows that is subharmonic. If is coordinated concave on , replace "'' by "'' above, we get that is superharmonic on .

We now give two version(s) of the Maximum (Minimum) Principle theorem using convexity on the coordinates.

Theorem 2.10.

Let be a coordinated convex (concave) function on . If is twice differentiable in and there is a point with , for all then is a constant function.

Proof.

By Proposition 2.9, we get that is sub(super)harmonic. Therefore, by Theorem 1.3 and the maximum principal the required result holds (see [14, page 264]).

Theorem 2.11.

Let and be two twice differentiable functions in . Assume that and are bounded real-valued functions defined on such that is coordinated convex and is coordinated concave. If for each point

(2.18)

then for all or and is harmonic.

Proof.

By Proposition 2.9, we get that is subharmonic and is superharmonic. Therefore, by Theorem 1.4 and using the maximum principal the required result holds, (see [14, page 264]).

Remark 2.12.

The above two results hold for log-convex functions on the coordinates, simply, replacing by , to obtain the results.

3. Some Inequalities and Applications

In the following we develop a Hadamard-type inequality for coordinated log-convex functions.

Corollary 3.1.

Suppose that is log-convex on the coordinates of , then

(3.1)

For a positive coordinated log-concave function , the inequalities are reversed.

Proof.

In Theorem 1.2, replace by and we get the required result.

Lemma 3.2.

For with , the function

(3.2)

is convex for all . Moreover,

(3.3)

for all .

Proof.

Since is twice differentiable for all with , we note that for all , , which shows that is increasing and thus is nonnegative which is equivalent to saying that is increasing and hence is convex. Now, using inequality (1.1), we get

(3.4)

which completes the proof.

Theorem 3.3.

Suppose that is log-convex on the coordinates of . Let

(3.5)

then the inequalities

(3.6)

hold, where is the Euler constant,

(3.7)

is the exponential integral function. For a coordinated log-concave function , the inequalities are reversed.

Proof.

Since is log-convex on the coordinates of , then

(3.8)

Integrating the previous inequality with respect to and on , we have,

(3.9)

Therefore, by (3.9) and for nonzero, positive , we have the following cases.

(1)If , the result is trivial.

(2)If , then

(3.10)

• (3) If , then

  

  (3.11)

• (4) If , then

  

  (3.12)

• (5) If , then

  

  (3.13)

• (6) If , then

  

  (3.14)

• (7) If , then

  

  (3.15)

• (8) If then

  

  (3.16)

Therefore, by Lemma 3.2, we deduce that

(3.17)

• (9) If , we have

(3.18)

which is difficult to evaluate because it depends on the values of and .

Remark 3.4.

The integrals in (3), (4), and (7) in the proof of Theorem 2.11 are evaluated using Maple Software.

Corollary 3.5.

In Theorem 3.3, if

(1), then

(3.19)

and for instance, if , we deduce

(3.20)

(2), then

(3.21)

and for instance, if , ? , we deduce

(3.22)

Proof.

Follows directly by applying inequality (1.4).