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

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

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

then

that is

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

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:

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:

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

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

where is given by (1.3) for convex function .

By inequality (1.11), we write

Corollary 1.6.

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

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

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

where

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:

Corollary 1.9.

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

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:

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:

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

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

where and for ,

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

Corollary 1.12.

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

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:

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:

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

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

for all positive integers .

We write (1.32) in the way that , where

for any positive integer .

The mixed symmetric means with positive weights related to

are defined as:

Corollary 1.15.

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

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:

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):

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

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

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

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

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

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

for every and every , .

Proposition 1.20.

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

Consider , defined as

and , defined as

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.

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

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

(b)The function is exponentially convex on .

Proof.

(i) Consider a function

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

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

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

Proof.

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

are convex.

We use in (1.43),

Similarly, by using in (1.43), we get

From (2.8) and (2.9), we get

Since , therefore

Hence, we have

Theorem 2.4.

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

provided that the denominators are non zero.

Proof.

Define in the way that

where and are as follow;

Now using Theorem 2.3 with , we have

Since , therefore (2.16) gives

Corollary 2.5.

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

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

3. Cauchy Mean

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

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

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

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

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

Proof.

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

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

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

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

The corresponding Cauchy means are given by

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

where and the limiting cases are as follows:

where .

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

Theorem 3.5.

Let such that , then

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

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.