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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 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.
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 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.
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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. 117-1170889-0888.
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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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DOI: https://doi.org/10.1155/2010/289730
Keywords
- Convex Function
- Similar Setting
- Positive Semidefinite
- Positive Weight
- Weight Version