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

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.

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 ,

(1.1)

and vice versa for .

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

Theorem 1.1.

For ,

(1.2)

where

(1.3)

For fixed let

(1.4)

denote two -tuples. Let

(1.5)

be their ordered components.

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

is said to majorize (or is said to be majorized by ), in symbol, , if

(1.6)

holds for and

(1.7)

Note that (1.6) is equivalent to

(1.8)

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.

Let be an interval in , and let be two -tuples such that , . Then

(1.9)

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.

Let be two decreasing real n-tuples, and let be a real -tuple such that

(1.10)

Then for every continuous convex function , one has

(1.11)

Definition 1.6.

A function is exponentially convex function if it is continuous and

(1.12)

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.

(ii) is continuous and

(1.13)

for every , every , and every , .

Corollary 1.8.

If is exponentially convex, then

(1.14)

for every and every , .

Corollary 1.9.

If is exponentially convex function, then is a -convex function in Jensens sense:

(1.15)

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.

Define the function

(2.1)

Then , that is, is convex for

Definition 2.2.

It is said that a positive function is -convex in the Jensen sense on some interval if

(2.2)

holds for every .

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

Lemma 2.3.

If is convex on an interval , then

(2.3)

holds for every , .

Theorem 2.4.

Let and be two positive -tuples, ,

(2.4)

and all 's and 's are not equal.

Then the following statements are valid.

(a)For every and , the matrix is a positive semidefinite matrix. Particularly

(2.5)

for

(b)The function is exponentially convex.

(c)The function is -convex on and the following inequality holds for :

(2.6)

Proof.

• (a) Consider the function

  

  (2.7)

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

We have

(2.8)

This shows that is a convex function for .

Using Theorem 1.3,

(2.9)

This implies that

(2.10)

or equivalently

(2.11)

From last inequality, it follows that the matrix is a positive semidefinite matrix, that is, (2.5) is valid.

• (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

(2.12)

This implies

(2.13)

that is, is a positive-valued function.

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

(2.14)

which is equivalent to (2.6).

Theorem 2.5.

Let be defined as in Theorem 2.4 and such that , , , and . Then

(2.15)

Proof.

For a convex function , a simple consequence of (2.3) is the following inequality [1, page 2]:

(2.16)

with . Since by Theorem 2.4(c) and is -convex, we can set in (2.16) , and . We get

(2.17)

from which (2.15) follows.

Theorem 2.6.

Let and be two positive decreasing -tuples, let be a real -tuple and let

(2.18)

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

Then the following statements are valid.

(a)For every and , the matrix is a positive semidefinite matrix. Particularly

(2.19)

for

(b)The function is exponentially convex.

(c)The function is -convex on and the following inequality holds for :

(2.20)

Proof.

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

Theorem 2.7.

Let be defined as in Theorem 2.6 and such that , , , and . Then

(2.21)

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.

Let , I interval in , be such that

(3.1)

Consider the functions , defined as

(3.2)

then for are convex.

Proof.

Since

(3.3)

that is, for are convex.

Denote

(3.4)

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

Theorem 3.2.

Let and be two positive -tuples, , all 's and 's are not equal, and with being defined as in (3.4), then there exists such that

(3.5)

Proof.

Since and is compact, then for Then by applying and defined in Lemma 3.1 for in Theorem 1.3, we have

(3.6)

that is,

(3.7)

(3.8)

By combining (3.7) and (3.8)

(3.9)

because , for by using Remark 1.4. Using the fact that for , there exists such that , we get (3.5).

Theorem 3.3.

Let and be two positive -tuples, , all 's and 's are not equa,and , with being defined as in (3.4), then there exists such that

(3.10)

provided that for every .

Proof.

Let a function be defined as

(3.11)

where and are defined as

(3.12)

Then, using Theorem 3.2 with , we have

(3.13)

Since

(3.14)

therefore, (3.13) gives

(3.15)

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

Corollary 3.4.

Let and be two positive -tuples such that and all 's and 's are not equal, then for there exists , with being defined as in (3.4), such that

(3.16)

Proof.

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

Remark 3.5.

Since the function is invertible, then from (3.16) we have

(3.17)

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

(3.18)

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 ,

(3.19)

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

(3.20)

Theorem 3.6.

Let such that , , then the following inequality is valid:

(3.21)

Proof.

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

Theorem 3.7.

Let and be two positive decreasing -tuples, let be a real -tuple such that conditions (1.10) are satisfied, is positive defined as in Theorem 2.6, and with being defined as in (3.4), then there exists such that

(3.22)

Proof.

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

Theorem 3.8.

Let and be two positive decreasing -tuples, be a real -tuple such that conditions (1.10) are satisfied, is positive defined as in Theorem 2.6 and , is defined as in (3.4). Then there exists such that

(3.23)

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 3.9.

Let and be two positive decreasing -tuples, let be a real -tuple such that conditions (1.10) are satisfied and is positive defined as in Theorem 2.6, then for there exists , with being defined as in (3.4), such that

(3.24)

Proof.

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

Remark 3.10.

Since the function is invertible, then from (3.24) we have

(3.25)

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

(3.26)

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

(3.27)

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

(3.28)

Theorem 3.11.

Let such that , , then the following inequality is valid:

(3.29)

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]).

is said to majorize , in symbol, , for if they are decreasing in and

(4.1)

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.

for iff they are decreasing in and

(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.

(a)If

(4.3)

(4.4)

hold, then for every continuous convex function , one has

(4.5)

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

Theorem 4.4.

Let and be two positive functions defined on an interval , decreasing in , ,

(4.6)

and is positive.

Then the following statements are valid.

(a)For every and , the matrix is a positive semidefinite matrix. Particularly

(4.7)

for

(b)The function is exponentially convex.

(c)The function is -convex on and the following inequality holds for

(4.8)

Proof.

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

Theorem 4.5.

Let be defined as in Theorem 4.4 and such that , , , and . Then

(4.9)

Proof.

Similar to the proof of Theorem 2.5.

Denote

(4.10)

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

Theorem 4.6.

Let and be two positive decreasing functions in such that , is positive defined as in Theorem 4.4, and , with being defined as in (4.10), then there exists such that

(4.11)

Proof.

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

Theorem 4.7.

Let and be two positive decreasing functions in such that , is positive defined as in Theorem 4.4, and , with being defined as in (4.10). Then there exists such that

(4.12)

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 4.8.

Let and be two positive decreasing functions in such that and is positive defined as in Theorem 4.4, then for there exists , with being defined as in (4.10), such that

(4.13)

Proof.

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

Remark 4.9.

Since the function is invertible, then from (4.13) we have

(4.14)

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

(4.15)

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

(4.16)

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

(4.17)

Theorem 4.10.

Let such that , , then the following inequality is valid:

(4.18)

Proof.

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

Theorem 4.11.

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

(4.19)

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

Then the following statements are valid.

(a)For every and , the matrix is a positive semidefinite matrix. Particularly

(4.20)

for

(b)The function is exponentially convex.

(c)The function is -convex on and the following inequality holds for

(4.21)

Proof.

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

Theorem 4.12.

Let be defined as in Theorem 4.11 and such that , , , and . Then

(4.22)

Proof.

Similar to the proof of Theorem 2.5.

Theorem 4.13.

Let and be positive, continuous, and increasing functions in such that conditions (4.3) and (4.4) are satisfied, is positive defined as in Theorem 4.11, with being defined as in (4.10), and be a function of bounded variation, then there exists such that

(4.23)

Proof.

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

Theorem 4.14.

Let and be positive, continuous and increasing functions in such that conditions (4.3) and (4.4) are satisfied, be a function of bounded variation, is positive defined as in Theorem 4.11, and , with being defined as in (4.10). Then there exists such that

(4.24)

provided that for every .

Proof.

Similar to the proof of Theorem 3.3.

Corollary 4.15.

Let and be positive, continuous, and increasing functions in such that conditions (4.3) and (4.4) be satisfied, is positive defined as in Theorem 4.11, and be a function of bounded variation, then for there exists , with being defined as in (4.10), such that

(4.25)

Proof.

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

Remark 4.16.

Since the function is invertible, then from (4.25) we have

(4.26)

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

(4.27)

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

(4.28)

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

(4.29)

Theorem 4.17.

Let such that , , then the following inequality is valid:

(4.30)

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 .