- Research Article
- Open Access

# Positive Semidefinite Matrices, Exponential Convexity for Majorization, and Related Cauchy Means

- M Anwar
^{1}Email author, - N Latif
^{2}and - J Pečarić
^{2, 3}

**2010**:728251

https://doi.org/10.1155/2010/728251

© M. Anwar et al. 2010

**Received:**26 October 2009**Accepted:**10 March 2010**Published:**16 March 2010

## Abstract

We prove positive semidefiniteness of matrices generated by differences deduced from majorization-type results which implies exponential convexity and log-convexity of these differences and also obtain Lyapunov's and Dresher's inequalities for these differences. We introduce new Cauchy means and show that these means are monotone.

## Keywords

- Convex Function
- Positive Function
- Related Result
- Simple Consequence
- Equivalent Condition

## 1. Introduction and Preliminaries

Let , denote two sequences of positive real numbers with . The well-known Jensen inequality for convex function [1, page 43] gives that, for or ,

and vice versa for .

In [2], the following generalization of this theorem is given.

Theorem 1.1.

For fixed let

denote two -tuples. Let

be their ordered components.

Definition 1.2 (see [1, page 319]).

for .

The following theorem is well-known as the majorization theorem and a convenient reference for its proof is given by Marshall and Olkin [3, page11] (see also [1, page 320]).

Theorem 1.3.

holds for every continuous convex function if and only if holds.

Remark 1.4 (see [4]).

If is a strictly convex function, then equality in (1.9) is valid iff .

The following theorem can be regarded as a generalization of Theorem 1.3 and is proved by Fuchs in [5] (see also [1, page 323]).

Theorem 1.5.

Definition 1.6.

for all and all choices and , such that , .

The following proposition is given in [6].

Proposition 1.7.

Let . The following propositions are equivalent.

(i) is exponentially convex.

for every , every , and every , .

Corollary 1.8.

for every and every , .

Corollary 1.9.

In this paper, we prove positive semidefiniteness of matrices generated by differences deduced from majorization-type results (1.9), (1.11), (4.2), and (4.5) which implies exponential convexity and log-convexity of these differences and also obtain Lyapunov's and Dresher's inequalities for these differences. In [7], new Cauchy means are introduced. By using these means, a generalization of (1.2) was given (see [7]). In the present paper, we give related results in discrete and indiscrete cases and some new means of the Cauchy type.

## 2. Main Results

Lemma 2.1.

Then , that is, is convex for

Definition 2.2.

holds for every .

The following lemma gives an equivalent condition for convexity of function [1, page 2].

Lemma 2.3.

holds for every , .

Theorem 2.4.

and all 's and 's are not equal.

Then the following statements are valid.

for

(b)The function is exponentially convex.

for , , , , where and is defined in (2.1).

We have

This shows that is a convex function for .

Using Theorem 1.3,

- (b)
Note that is continuous for . Then by using Proposition 1.7, we get exponentially convexity of the function .

- (c)
Since is continuous and strictly convex function for and all 's and 's are not equal, therefore by Theorem 1.3 with we have

that is, is a positive-valued function.

A simple consequence of Corollary 1.9 is that is -convex; then by definition

which is equivalent to (2.6).

Theorem 2.5.

Proof.

from which (2.15) follows.

Theorem 2.6.

such that conditions (1.10) are satisfied and is positive.

Then the following statements are valid.

for

(b)The function is exponentially convex.

Proof.

As in the proof of Theorem 2.4, we use Theorem 1.5 instead of Theorem 1.3.

Theorem 2.7.

Proof.

Similar to the proof of Theorem 2.5.

## 3. Cauchy Means

Let us note that (2.15) and (2.21) have the form of some known inequalities between means (e.g., Stolarsky means, Gini means, etc). Here we will prove that expressions on both sides of (2.15) and (2.21) are also means.

Lemma 3.1.

then for are convex.

Proof.

that is, for are convex.

Denote

In the above expression, and are the minimums of and , respectively. Similarly, and are the maximums of and respectively.

Theorem 3.2.

Proof.

Theorem 3.3.

provided that for every .

Proof.

After putting values, we get (3.10). The denominator of left-hand side is nonzero by using in Theorem 3.2.

Corollary 3.4.

Proof.

Setting and , in (3.10), we get (3.16).

Remark 3.5.

In fact, similar result can also be given for (3.10). Namely, suppose that has inverse function. Then from (3.10), we have

So, we have that the expression on the right-hand side of (3.18) is also a mean. By the inequality (3.17), we can consider for positive -tuples and such that ,

for , as means in broader sense. Moreover we can extend these means in other cases. So passing to the limit, we have

Theorem 3.6.

Proof.

Since is log-convex, therefore by (2.15) we get (3.21)

Theorem 3.7.

Proof.

As in the proof of Theorem 3.2, we use Theorem 1.5 instead of Theorem 1.3.

Theorem 3.8.

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 3.9.

Proof.

Setting and , in (3.23), we get (3.24).

Remark 3.10.

In fact, similar result can also be given for (3.23). Namely, suppose that has inverse function. Then from (3.23), we have

So, we have that the expression on the right-hand side of (3.26) is also a mean. By the inequality (3.25), we can consider for positive -tuples and such that conditions (1.10) are satisfied, and

for , as means in broader sense. Moreover we can extend these means in other cases. So passing to the limit, we have

Theorem 3.11.

Proof.

Since is -convex, therefore by (2.21) we get (3.29)

## 4. Some Related Results

Let , be real valued functions defined on an interval such that , both exist for all .

Definition 4.1 (see [1, page 324]).

and equality in (4.1) holds for .

The following theorem can be regarded as a majorization theorem in integral case [1, page 325].

Theorem 4.2.

holds for every that is continuous, and convex in such that the integrals exist.

The following theorem is a simple consequence of Theorem in [8] (see also [1, page 328]):

Theorem 4.3.

Let , and are continuous and increasing and let be a function of bounded variation.

(b)If (4.3) holds, then (4.5) holds for every continuous increasing convex function .

Theorem 4.4.

and is positive.

Then the following statements are valid.

for

(b)The function is exponentially convex.

Proof.

As in the proof of Theorem 2.4, we use Theorem 4.2 instead of Theorem 1.3.

Theorem 4.5.

Proof.

Similar to the proof of Theorem 2.5.

Denote

In the above expression, and are the minimums of and , respectively. Similarly, and are the maximums of and , respectively.

Theorem 4.6.

Proof.

As in the proof of Theorem 3.2, we use Theorem 4.2 instead of Theorem 1.3.

Theorem 4.7.

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 4.8.

Proof.

Set and , in (4.12), we get (4.13).

Remark 4.9.

In fact, similar result can also be given for (4.12). Namely, suppose that has inverse function. Then from (4.12), we have

So, we have that the expression on the right-hand side of (4.15) is also a mean. By the inequality (4.14), we can consider for positive functions and such that , and

for , as means in broader sense. Moreover we can extend these means in other cases. So passing to the limit, we have

Theorem 4.10.

Proof.

Since is -convex, therefore by (4.9) we get (4.18).

Theorem 4.11.

such that conditions (4.3) and (4.4) are satisfied and is positive.

Then the following statements are valid.

for

(b)The function is exponentially convex.

Proof.

As in the proof of Theorem 2.4, we use Theorem 4.3 instead of Theorem 1.3.

Theorem 4.12.

Proof.

Similar to the proof of Theorem 2.5.

Theorem 4.13.

Proof.

As in the proof of Theorem 3.2, we use Theorem 4.3 instead of Theorem 1.3.

Theorem 4.14.

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 4.15.

Proof.

Setting and , in (4.24), we get (4.25).

Remark 4.16.

In fact, similar result can also be given for (4.24). Namely, suppose that has inverse function. Then from (4.24), we have

So, we have that the expression on the right-hand side of (4.27) is also a mean. By the inequality (4.26), we can consider for positive functions and such that conditions (4.3) and (4.4) are satisfied, and

Theorem 4.17.

Proof.

Since is -convex, therefore by (4.22) we get (4.30)

Remark 4.18.

Let such that and . If we substitute in Theorem 2.6 , we get (1.2). In fact in such results we have that is monotonic -tuple. But since the weights are positive, our results are also valid for arbitrary .

## Declarations

### Acknowledgments

This research work is funded by Higher Education Commission of Pakistan. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888. The authors thank I. Olkin for many valuable suggestions.

## Authors’ Affiliations

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## Copyright

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