Improvement and Reversion of Slater's Inequality and Related Results

Abstract

We use an inequality given by Matić and Pečarić (2000) and obtain improvement and reverse of Slater's and related inequalities.

1. Introduction

In 1981 Slater has proved an interesting companion inequality to Jensen's inequality [1].

Theorem 1.1.

Suppose that is increasing convex function on interval , for (where is the interior of the interval ) and for with , if , then

(1.1)

When is strictly convex on , inequality (1.1) becomes equality if and only if for some and for all with .

It was noted in [2] that by using the same proof the following generalization of Slater's inequality (1981) can be given.

Theorem 1.2.

Suppose that is convex function on interval , for (where is the interior of the interval ) and for with . Let

(1.2)

then inequality (1.1) holds.

When is strictly convex on , inequality (1.1) becomes equality if and only if for some and for all with .

Remark 1.3.

For multidimensional version of Theorem 1.2 see [3].

Another companion inequality to Jensen's inequality is a converse proved by Dragomir and Goh in [4].

Theorem 1.4.

Let be differentiable convex function defined on interval . If are arbitrary members and with and let

(1.3)

Then the inequalities

(1.4)

hold.

In the case when is strictly convex, one has equalities in (1.4) if and only if there is some such that holds for all with

Matić and Pečarić in [5] proved more general inequality from which (1.1) and (1.4) can be obtained as special cases.

Theorem 1.5.

Let be differentiable convex function defined on interval and let and be stated as in Theorem 1.4. If is arbitrary chosen number, then one has

(1.5)

Also, when is strictly convex, one has equality in (1.5) if and only if holds for all with .

Remark 1.6.

If and are stated as in Theorem 1.4 and we let , also if , then by setting in (1.5), we get Slater's inequality (1.1) and similarly by setting in (1.5), we get (1.4).

The following refinement of (1.4) is also valid [5].

Theorem 1.7.

Let be strictly convex differentiable function defined on interval and let and be stated as in Theorem 1.4 and , then the inequalities

(1.6)

(1.7)

hold.

The equalities hold in (1.6) and in (1.7) if and only if

Remark 1.8.

In [6] Dragomir has also proved Theorem 1.7.

In this paper, we use an inequality given in [5] and derive two mean value theorems, exponential convexity, log-convexity, and Cauchy means. As applications, such results are also deduce for related inequality. We use some log-convexity criterion and prove improvement and reverse of Slater's and related inequalities. We also prove some determinantal inequalities.

2. Mean Value Theorems

Theorem 2.1.

Let , where is closed interval in , and let , , with and . Then there exists such that

(2.1)

Proof.

Since is continuous on , for , where and .

Consider the functions , defined as

(2.2)

Since

(2.3)

for are convex.

Now by applying for in inequality (1.5), we have

(2.4)

From (2.4) we get

(2.5)

and similarly by applying for in (1.5), we get

(2.6)

Since

(2.7)

by combining (2.5) and (2.6), we have

(2.8)

Now using the fact that for there exists such that , we get (2.1).

Corollary 2.2.

Let , where is closed interval in , and let , and be stated as in Theorem 1.4 with and . Then there exists such that

(2.9)

Proof.

By setting in Theorem 2.1, we get (2.9).

Theorem 2.3.

Let , where is closed interval in , and let , and with . Then there exists such that

(2.10)

provided that the denominators are nonzero.

Proof.

Let the function be defined by

(2.11)

where and are defined as

(2.12)

Then, using Theorem 2.1 with , we have

(2.13)

because .

Since as and , therefore, (2.13) gives us

(2.14)

After putting the values of and , we get (2.10).

Corollary 2.4.

Let , where is closed interval in , and , and let , with . Then there exists such that

(2.15)

provided that the denominators are nonzero.

Proof.

By setting in Theorem 2.3, we get (2.15).

Corollary 2.5.

Let with and , . Then for , , there exists , where is positive closed interval, such that

(2.16)

Proof.

By setting and , , in Theorem 2.3, we get (2.16).

Corollary 2.6.

Let , , and with . Then for , , there exists , where is positive closed interval, such that

(2.17)

Proof.

By setting and , , in (2.15), we get (2.17).

Remark 2.7.

Note that we can consider the interval , where ,

Since the function with is invertible, then from (2.16) we have

(2.18)

We will say that the expression in the middle is a mean of .

From (2.17) we have

(2.19)

The expression in the middle of (2.19) is a mean of .

In fact similar results can also be given for (2.10) and (2.15). Namely, suppose that has inverse function, then from (2.10) and (2.15) we have

(2.20)

So, we have that the expression on the right-hand side of (2.20) is also means.

3. Improvements and Related Results

Definition 3.1 (see [7, page 2]).

A function is convex if

(3.1)

holds for every ,

Lemma 3.2 (see [8]).

Let one define the function

(3.2)

Then , that is, is convex for .

Definition 3.3 (see [9]).

A function is exponentially convex if it is continuous and

(3.3)

for all , and , such that or equivalently

(3.4)

Corollary 3.4 (see [9]).

If is exponentially convex function, then

(3.5)

for every

Corollary 3.5 (see [9]).

If is exponentially convex function, then is a log-convex function that is

(3.6)

Theorem 3.6.

Let , . Consider to be defined by

(3.7)

Then

(i)for every and for every , the matrix is a positive semidefinite matrix; particularly

(3.8)

(ii)the function is exponentially convex;

(iii)if , then the function is log-convex, that is, for , one has

(3.9)

Proof.

Let us consider the function defined by

(3.10)

where for all

Then we have

(3.11)

Therefore, is convex function for . Using in inequality (1.5), we get

(3.12)

so the matrix is positive semi-definite.

Since and , so is continuous for all and we have exponentially convexity of the function .

• (iii) Let , then by Corollary 3.5 we have that is log-convex, that is, is convex, and by (3.1) for and taking , we get

  

  (3.13)

which is equivalent to (3.9).

Corollary 3.7.

Let , and . Consider to be defined by

(3.14)

Then

(i)for every and for every , the matrix is a positive semi-definite matrix. Particularly

(3.15)

(ii)the function is exponentially convex;

(iii)if , then the function is log-convex, that is, for , one has

(3.16)

Proof.

To get the required results, set in Theorem 3.6.

Let be positive n-tuple and positive real numbers, and let . Let denote the power mean of order , defined by

(3.17)

Let us note that .

By (2.18) we can give the following definition of Cauchy means.

Let with , is positive closed interval, and ,

(3.18)

for are means of . Moreover we can extend these means to the other cases.

So by limit we have

(3.19)

where .

Theorem 3.8.

Let such that , then the following inequality is valid:

(3.20)

Proof.

For convex function it holds that ([7, page 2])

(3.21)

with , , , Since by Theorem 3.6, is log-convex, we can set in (3.21): , , , , and , then we get

(3.22)

From (3.22) we get (3.20) for and .

For and we have limiting case.

Similarly by (2.19) we can give the following definition of Cauchy type means.

Let with , is positive closed interval, and

(3.23)

for are means of . Moreover we can extend these means to the other cases.

So by limit we have

(3.24)

where .

Theorem 3.9.

Let such that , then the following inequality is valid:

(3.25)

Proof.

The proof is similar to the proof of Theorem 3.8.

Let be stated as above, define as

(3.26)

The following improvement and reverse of Slater's inequality are valid.

Theorem 3.10.

Let , . Let be defined by

(3.27)

Then

• (i)
  

  (3.28)

for and .

• (ii)
  

  (3.29)

for .

where,

(3.30)

Proof.

• (i) By setting in (3.7), becomes and for , setting in (3.9), we get

  

  (3.31)

that is,

(3.32)

From (3.32) we get (3.28), and similarly for (3.9) becomes

(3.33)

by the same process we can get (3.28).

• (ii) For (3.9) becomes

  

  (3.34)

setting in (3.34), we get (3.29).

Theorem 3.11.

Let , .

Then for every and for every , the matrices , are positive semi-definite matrices. Particularly

(3.35)

(3.36)

where is defined by (3.30).

Proof.

By setting and in Theorem 3.6(i), we get the required results.

Remark 3.12.

We note that . So by setting in (3.35), we have special case of (3.28) for ,  and if and for ,  and if . Similarly by setting in (3.36), we have special case of (3.29) for if and for if .

Let be stated as above, define as

(3.37)

The following improvement and reverse of inequality (1.6) are also valid.

Theorem 3.13.

Let for all , . Let be defined by

(3.38)

Then

• (i)
  

  (3.39)

for and .

• (ii)
  

  (3.40)

for ,

where

(3.41)

Proof.

• (i) By setting in (3.9), we get (3.39) for , and similarly we can get (3.39) for the case .

• (ii) For (3.9) becomes

  

  (3.42)

setting in (3.42), we get (3.40).

Theorem 3.14.

Let , .

Then for every and for every , the matrices , are positive semi-definite matrices. Particularly

(3.43)

(3.44)

where is defined by (3.41).

Proof.

By setting and in Theorem 3.6(i), we get the required results.

Remark 3.15.

We note that . So by setting in (3.43), we have special case of (3.39) for if and for , and   if . Similarly by setting in (3.44), we have special case of (3.40) for ,  and   if and for , and  if .