Differences of Weighted Mixed Symmetric Means and Related Results

Abstract

Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.

1. Introduction and Preliminary Results

For , let and be positive -tuples such that . We define power means of order , as follows:

(1.1)

We introduce the mixed symmetric means with positive weights as follows:

(1.2)

We obtain the monotonicity of these means as a consequence of the following improvement of Jensen's inequality [1].

Theorem 1.1.

Let , , be a positive -tuple such that . Also let be a convex function and

(1.3)

then

(1.4)

that is

(1.5)

If is a concave function, then the inequality (1.4) is reversed.

Corollary 1.2.

Let such that , and let and be positive -tuples such that , then, we have

(1.6)

(1.7)

Proof.

Let such that , if , then we set , in (1.4) and raising the power , we get (1.6). Similarly we set , in (1.4) and raising the power , we get (1.7).

When or , we get the required results by taking limit.

Let be an interval, , be positive -tuples such that . Also let be continuous and strictly monotonic functions. We define the quasiarithmetic means with respect to (1.3) as follows:

(1.8)

where is the convex function.

We obtain generalized means by setting , and applying to (1.3).

Corollary 1.3.

By similar setting in (1.4), one gets the monotonicity of generalized means as follows:

(1.9)

where is convex and is increasing, or is concave and is decreasing;

(1.10)

where is convex and is decreasing, or is concave and is increasing.

Remark 1.4.

In fact Corollaries 1.2 and 1.3 are weighted versions of results in [2].

The inequality of Popoviciu as given by Vasić and Stanković in [3] (see also [4, page 173]) can be written in the following form:

Theorem 1.5.

Let the conditions of Theorem 1.1 be satisfied for , , . Then

(1.11)

where is given by (1.3) for convex function .

By inequality (1.11), we write

(1.12)

Corollary 1.6.

Let such that , and let and be positive -tuples such that . Then, we have

(1.13)

(1.14)

Proof.

Let such that , if , then we set , in (1.11) to obtain (1.13) and we set , in (1.11) to obtain (1.14).

When or , we get the required results by taking limit.

Corollary 1.7.

We set and the convex function in (1.11) to get

(1.15)

The following result is valid [5, page 8].

Theorem 1.8.

Let be a convex function defined on an interval , , be positive -tuples such that and . Then

(1.16)

where

(1.17)

If is a concave function then the inequality (1.16) is reversed.

We introduce the mixed symmetric means with positive weights related to (1.17) as follows:

(1.18)

Corollary 1.9.

Let such that , and let and be positive -tuples such that . Then, we have

(1.19)

(1.20)

Proof.

Let such that , if , then we set , in (1.16) and raising the power , we get (1.19). Similarly we set , in (1.16) and raising the power , we get (1.20).

When or , we get the required results by taking limit.

We define the quasiarithmetic means with respect to (1.17) as follows:

(1.21)

where is the convex function.

We obtain these generalized means by setting , and applying to (1.17).

Corollary 1.10.

By similar setting in (1.16), we get the monotonicity of these generalized means as follows:

(1.22)

where is convex and is increasing, or is concave and is decreasing;

(1.23)

where is convex and is decreasing, or is concave and is increasing.

The following result is given in [4, page 90].

Theorem 1.11.

Let be a real linear space, a non empty convex set in , a convex function, and also let be positive -tuples such that and . Then

(1.24)

where and for ,

(1.25)

The mixed symmetric means with positive weights related to (1.25) are

(1.26)

Corollary 1.12.

Let such that , and let and be positive -tuples such that . Then, we have

(1.27)

(1.28)

Proof.

Let such that , if , then we set , in (1.24) and raising the power , we get (1.27). Similarly we set , in (1.25) and raising the power , we get (1.28).

When or , we get the required results by taking limit.

We define the quasiarithmetic means with respect to (1.25) as follows:

(1.29)

where is the convex function.

We obtain these generalized means be setting , and applying to (1.25).

Corollary 1.13.

By similar setting in (1.24), we get the monotonicity of generalized means as follows:

(1.30)

where is convex and is increasing, or is concave and is decreasing;

(1.31)

where is convex and is decreasing, or is concave and is increasing.

The following result is given at [4, page 97].

Theorem 1.14.

Let , be a convex function, be an increasing function on such that , and be -integrable on . Then

(1.32)

for all positive integers .

We write (1.32) in the way that , where

(1.33)

for any positive integer .

The mixed symmetric means with positive weights related to

(1.34)

are defined as:

(1.35)

Corollary 1.15.

Let such that , and let and be positive -tuples such that . Then, we have

(1.36)

(1.37)

Proof.

Let such that , if , then we set , in (1.32) and raising the power , we get (1.36). Similarly we set , in (1.32) and raising the power , we get (1.37).

When or , we get the required results by taking limit.

We define the quasiarithmetic means with respect to (1.32) as follows:

(1.38)

where is the convex function.

We obtain these generalized means by setting , and applying to (1.34).

Corollary 1.16.

By similar setting in (1.32), we get the monotonicity of generalized means, given in (1.38):

(1.39)

where is convex and is increasing, or is concave and is decreasing;

(1.40)

where is convex and is decreasing, or is concave and is increasing.

Remark 1.17.

In fact unweighted version of these results were proved in [6], but in Remark 2.14 from [6], it is written that the same is valid for weighted case.

For convex function , we define

(1.41)

from (1.4), (1.16), and (1.24), in the way that

(1.42)

combining (1.42) with (1.12) and (1.33), we have

(1.43)

for any convex function .

The exponentially convex functions are defined in [7] as follows.

Definition 1.18.

A function is exponentially convex if it is continuous and

(1.44)

for all and all choices and , .

We also quote here a useful propositions from [7].

Proposition 1.19.

Let be a function, then following statements are equivalent;

(i) is exponentially convex.

(ii) is continuous and

(1.45)

for every and every , .

Proposition 1.20.

If is an exponentially convex function then is a log-convex function.

Consider , defined as

(1.46)

and , defined as

(1.47)

It is easy to see that both and are convex.

In this paper we prove the exponential convexity of (1.43) for convex functions defined in (1.46) and (1.47) and mean value theorems for the differences given in (1.43). We also define the corresponding means of Cauchy type and establish their monotonicity.

2. Main Result

The following theorems are the generalizations of results given in [6].

Theorem 2.1.

• (i) Let the conditions of Theorem 1.1 be satisfied. Consider

  

  (2.1)

where is obtained by replacing convex function with for , in . Then the following statements are valid.

(a)For every and , the matrix is a positive semidefinite matrix. Particularly

(2.2)

(b)The function is exponentially convex on .

Proof.

• (i) Consider a function

  

  (2.3)

for , , , and are not simultaneously zero and . We have

(2.4)

It follows that is a convex function. By taking in (1.43), we have

(2.5)

This means that the matrix is a positive semidefinite, that is, (2.2) is valid.

• (ii) It was proved in [6] that is continuous for . By using Proposition 1.19, we get exponential convexity of the function .

Theorem 2.2.

Theorem 2.1 is still valid for convex functions .

Theorem 2.3.

Let and be positive integers such that and let , , then there exists such that

(2.6)

Proof.

Since therefore there exist real numbers and . It is easy to show that the functions , defined as

(2.7)

are convex.

We use in (1.43),

(2.8)

Similarly, by using in (1.43), we get

(2.9)

From (2.8) and (2.9), we get

(2.10)

Since , therefore

(2.11)

Hence, we have

(2.12)

Theorem 2.4.

Let and be positive integer such that and , then there exists such that

(2.13)

provided that the denominators are non zero.

Proof.

Define in the way that

(2.14)

where and are as follow;

(2.15)

Now using Theorem 2.3 with , we have

(2.16)

Since , therefore (2.16) gives

(2.17)

Corollary 2.5.

Let and be positive -tuples, then for distinct real numbers and , different from zero and 1, there exists , such that

(2.18)

Proof.

Taking and , in (2.13), for distinct real numbers and , different from zero and 1, we obtain (2.18).

Remark 2.6.

Since the function , is invertible, then from (2.18), we get

(2.19)

3. Cauchy Mean

In fact, similar result can also be find for (2.13). Suppose that has inverse function. Then (2.13) gives

(3.1)

We have that the expression on the right hand side of above, is also a mean. We define Cauchy means

(3.2)

Also, we have continuous extensions of these means in other cases. Therefore by limit, we have the following:

(3.3)

The following lemma gives an equivalent definition of the convex function [4, page 2].

Lemma 3.1.

Let be a convex function defined on an interval and . Then

(3.4)

Now, we deduce the monotonicity of means given in (3.2) in the form of Dresher's inequality, as follows.

Theorem 3.2.

Let be given as in (3.2) and such that , , then

(3.5)

Proof.

By Proposition 1.20   is -convex. We set in Lemma 3.1 and get

(3.6)

This together with (2.1) follows (3.5).

Corollary 3.3.

Let and be positive -tuples, then for distinct real numbers , , and , all are different from zero and 1, there exists , such that

(3.7)

Proof.

Set and , then taking in (2.13), we get (3.7) by the virtue of (1.2), (1.18), (1.26) and (1.35) for non zero, distinct real numbers , and .

Remark 3.4.

Since the function is invertible, then from (3.7) we get

(3.8)

where , , and are non zero, distinct real numbers.

The corresponding Cauchy means are given by

(3.9)

where , , and are non zero, distinct real numbers. We write (3.9) as

(3.10)

where and the limiting cases are as follows:

(3.11)

where .

Now, we give the monotonicity of new means given in (3.10), as follows:

Theorem 3.5.

Let such that , then

(3.12)

where is given in (3.10).

Proof.

We take as defined in Theorem 2.1. are log-convex by Proposition 1.20, therefore by Lemma 3.1 for , , , we get

(3.13)

For , we set , , , , such that , , in (2.1) to obtain (3.12) with the help of (3.13).

Similarly for , we set , , , , such that , , in (2.1) and get (3.12) again, by the virtue of (3.13).

In the case , since for is continuous therefore We get required result by taking limit.