- Research Article
- Open Access

# Improvement and Reversion of Slater's Inequality and Related Results

- M Adil Khan
^{1}Email author and - JE Pečarić
^{1, 2}

**2010**:646034

https://doi.org/10.1155/2010/646034

© M. Adil Khan and J. E. Pečarić. 2010

**Received:**6 March 2010**Accepted:**2 June 2010**Published:**15 June 2010

## Abstract

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

## Keywords

- Convex Function
- Relate Result
- Positive Real Number
- Related Inequality
- Interesting Companion

## 1. Introduction

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

Theorem 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.

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.

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.

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.

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.

Proof.

Since is continuous on , for , where and .

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

Corollary 2.2.

Proof.

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

Theorem 2.3.

provided that the denominators are nonzero.

Proof.

because .

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

Corollary 2.4.

provided that the denominators are nonzero.

Proof.

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

Corollary 2.5.

Proof.

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

Corollary 2.6.

Proof.

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

Remark 2.7.

Note that we can consider the interval , where ,

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

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

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]).

holds for every ,

Lemma 3.2 (see [8]).

Then , that is, is convex for .

Definition 3.3 (see [9]).

Corollary 3.4 (see [9]).

for every

Corollary 3.5 (see [9]).

Theorem 3.6.

Then

(ii)the function is exponentially convex;

Proof.

where for all

so the matrix is positive semi-definite.

- (iii)

which is equivalent to (3.9).

Corollary 3.7.

Then

(ii)the function is exponentially convex;

Proof.

To get the required results, set in Theorem 3.6.

Let us note that .

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

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

where .

Theorem 3.8.

Proof.

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.

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

where .

Theorem 3.9.

Proof.

The proof is similar to the proof of Theorem 3.8.

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

Theorem 3.10.

for .

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

Theorem 3.11.

Let , .

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 .

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

Theorem 3.13.

for ,

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

- (ii)

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

Theorem 3.14.

Let , .

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 .

## Declarations

### Acknowledgments

The research of the first and second authors was 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 117-1170889-0888.

## Authors’ Affiliations

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## Copyright

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