# Refinements of Results about Weighted Mixed Symmetric Means and Related Cauchy Means

- László Horváth
^{1}, - KhuramAli Khan
^{2, 3}Email author and - J Pečarić
^{2, 4}

**2011**:350973

https://doi.org/10.1155/2011/350973

© László Horváth et al. 2011

**Received: **26 November 2010

**Accepted: **23 February 2011

**Published: **14 March 2011

## Abstract

A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.

## 1. Introduction and Preliminary Results

A new refinement of the discrete Jensen inequality is given in [1]. The following notations are also introduced in [1].

It is easy to observe from the construction of the functions and that they do not depend essentially on , so we can write for short for , and so on.

By (1.6), and this implies that for . From (1.6), again, we have .

We need the following hypotheses.

(H_{2})Let
and
be positive
-tuples such that
.

(H_{3}) Let
be an interval, let
, let
be a positive
-tuples such that
, and let
,
be continuous and strictly monotone functions.

(H_{4}) Let
be an interval, let
, and let
be positive
-tuples such that
. Further, let
be a convex function.

We deduce the monotonicity of these means from the following refinement of the discrete Jensen inequality in [1].

Theorem 1.1.

where the numbers are defined in (1.10) and (1.11). If is a concave function, then the inequalities in (1.16) are reversed.

Corollary 1.2.

Proof.

Assume . To obtain (1.18), we can apply Theorem 1.1 to the function and the -tuples to get the analogue of (1.16) and to raise the power . Equation (1.19) can be proved in a similar way by using and and raising the power .

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

_{1}) and (H

_{3}). Then, we define the quasiarithmetic means with respect to (1.10) and (1.11) as follows:

The monotonicity of these generalized means is obtained in the next corollary.

Corollary 1.3.

if either is convex and is decreasing or is concave and is increasing.

Proof.

First, we can apply Theorem 1.1 to the function and the -tuples , then we can apply to the inequality coming from (1.16). This gives (1.23). A similar argument gives (1.24): and can be used.

Throughout Examples 1.4-1.5, 1.9–1.12, which are based on examples in [1], the conditions (H_{2}), in the mixed symmetric means, and (H_{3}), in the quasiarithmetic means, will be assumed.

Example 1.4.

Example 1.5.

The following result is also given in [1].

Theorem 1.6.

If is a concave function then the inequalities (1.38) are reversed.

_{1}) and (H

_{2}), and suppose for any . In this case, the power means of order corresponding to has the form

Corollary 1.7.

Proof.

The proof comes from Corollary 1.2.

_{1}) and (H

_{3}), and suppose for any . We define for the quasiarithmetic means with respect to (1.37) as follows:

Corollary 1.8.

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

Proof.

The proof is a consequence of Corollary 1.3.

Example 1.9.

Equation (1.48) is a weighted mixed symmetric mean and (1.49) is a generalized mean, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.2 and 1.3 given in [2].

Example 1.10.

Equation (1.54) represents weighted mixed symmetric means, and (1.55) defines generalized means, as given in [2]. Therefore, Corollaries 1.7 and 1.8 are more general than the Corollaries 1.9 and 1.10 given in [2].

Example 1.11.

respectively.

Example 1.12.

respectively.

## 2. Main Results

From now on, (H_{1}) and (H_{4}) are assumed if we consider
, further, the hypothesis
for any
is also assumed if we consider
. The numbers
are generated by concrete examples, and (H_{4}) is assumed.

We need the following subclass of convex functions (see [3]).

Definition 2.1.

We quote here useful propositions from [3].

Proposition 2.2.

Let be a function. Then, the following statements are equivalent

Proposition 2.3.

First, we introduce a special class of functions.

(H_{5}) Let
be a set of twice differentiable convex functions such that the function
is exponentially convex for every fixed
.

It is easy to see that the sets of functions
and
satisfy (H_{5}).

In this paper we prove the exponential convexity of the functions , and we give mean value theorems for . We also define the respective means of Cauchy type and study their monotonicity. The results for are special cases of the results for , and the results for are special cases of results for . Especially, the results for are also given in [2].

Theorem 2.4.

Assume (H_{5}), and suppose that the functions
are continuous. The following statements hold for
.

(b)The function is exponentially convex.

Proof.

This means that the matrix is positive semidefinite, that is, (2.9) is valid.

(b)It is assumed that the function is continuous. By using Poposition 2.2 and (a), we get the exponential convexity of the function .

Since the functions are continuous , we have the following.

Corollary 2.5.

The function are exponentially convex. This remains valid if we replace by in (2.8).

## 3. Cauchy Means

In this section, first, we are interested in mean value theorems.

Theorem 3.1.

Theorem 3.2.

provided that the denominators are nonzero.

The idea of the proofs of Theorems 3.1 and 3.2 is the same as the proofs of Theorems 2.3 and 2.4 in [2].

Corollary 3.3.

Proof.

Theorem 3.2 can be applied.

Remark 3.4.

Suppose the conditions of Corollary 3.3 are satisfied.

(a)Since the function , is invertible, then we get, from (3.3), that for

Corollary 3.5.

_{1}) and (H

_{2}), and suppose . In (3.6), it is also supposed that for any . Then, for distinct real numbers , , and , all are different from 0 and 1, there exists , such that

Proof.

We can apply Theorem 3.2 to the functions , , , and , and the -tuples .

Remark 3.6.

Suppose the conditions of Corollary 3.5 are satisfied.

(a)Since the function is invertible, then we get, from (3.5) and (3.6) that

(b)As in Remark 3.4 (b), the expressions in (3.7) define means.

Now, we deduce the monotonicity of these means in the form of Dresher's inequality as follows.

Theorem 3.7.

Proof.

This gives (3.10) if and . The other cases come from this by taking limit.

Now, we give the monotonicity of these new means.

Theorem 3.8.

Proof.

## Funding

This research was partially funded by the Higher Education Commission, Pakistan.

## Declarations

### Acknowledgments

The research of L. Horváth was supported by the Hungarian National Foundations for Scientific Research, Grant no. K73274, and that of J. Pecărić was supported by the Croatian Ministry of Science, Education, and Sports under the research Grant 117-1170889-0888.

## Authors’ Affiliations

## References

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## Copyright

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