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Differences of Weighted Mixed Symmetric Means and Related Results
Journal of Inequalities and Applications volume 2010, Article number: 289730 (2010)
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:
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 logconvex 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.
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
(b)The function is exponentially convex on .
Proof.

(i)
Consider a function
(2.3)
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 logconvex 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.
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Acknowledgments
This research was partially funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant no. 11711708890888.
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Khan, K., Pečarić, J. & Perić, I. Differences of Weighted Mixed Symmetric Means and Related Results. J Inequal Appl 2010, 289730 (2010). https://doi.org/10.1155/2010/289730
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Keywords
 Convex Function
 Similar Setting
 Positive Semidefinite
 Positive Weight
 Weight Version