# Generalization of an Inequality for Integral Transforms with Kernel and Related Results

- Sajid Iqbal
^{1}Email author, - J Pečarić
^{1, 2}and - Yong Zhou
^{3}

**2010**:948430

https://doi.org/10.1155/2010/948430

© Sajid Iqbal et al. 2010

**Received: **27 March 2010

**Accepted: **27 October 2010

**Published: **31 October 2010

## Abstract

We establish a generalization of the inequality introduced by Mitrinović and Pečarić in 1988. We prove mean value theorems of Cauchy type for that new inequality by taking its difference. Furthermore, we prove the positive semidefiniteness of the matrices generated by the difference of the inequality which implies the exponential convexity and logarithmic convexity. Finally, we define new means of Cauchy type and prove the monotonicity of these means.

## Keywords

## 1. Introduction

for any continuous function on . Throughout the paper, it is assumed that all integrals under consideration exist and that they are finite.

The following theorem is given in [1] (see also [2, page 235]).

Theorem 1.1.

The following definition is equivalent to the definition of convex functions.

Definition 1.2 (see [2]).

Let us recall the following definition.

Definition 1.3 (see [3, page 373]).

The following proposition is useful to prove the exponential convexity.

Proposition 1.4 (see [4]).

Let . The following statements are equivalent.

Corollary 1.5.

This paper is organized in this manner. In Section 2, we give the generalization of Mitrinović-Pečarić inequality and prove the mean value theorems of Cauchy type. We also introduce the new type of Cauchy means. In Section 3, we give the proof of positive semidefiniteness of matrices generated by the difference of that inequality obtained from the generalization of Mitrinović-Pečarić inequality and also discuss the exponential convexity. At the end, we prove the monotonicity of the means.

## 2. Main Results

Theorem 2.1.

Proof.

Remark 2.2.

If is strictly convex on and is nonconstant, then the inequality in (2.1) is strict.

Remark 2.3.

From (2.6) and (2.7), we get (1.2).

Corollary 2.4.

Let be space of all absolutely continuous functions on . By , we denote the space of all functions with .

*α*for a function is defined by

where ; the notation of stands for the largest integer not greater than .

Corollary 2.5.

Let be the space of all functions integrable on . For , we say that has an fractional derivative in if and only if for , , and .

The next lemma is very useful to give the upcoming corollary [6] (see also [5, p. 449]).

Lemma 2.6.

Corollary 2.7.

Lemma 2.8.

Proof.

Theorem 2.9.

Proof.

This is the claim of the theorem.

Let us note that a generalized mean value Theorem 2.9 for fractional derivative was given in [7]. Here we will give some related results as consequences of Theorem 2.9.

Corollary 2.10.

Corollary 2.11.

Corollary 2.12.

Theorem 2.13.

It is provided that denominators are not equal to zero.

Proof.

This is the claim of the theorem.

Let us note that a generalized Cauchy mean-valued theorem for fractional derivative was given in [8]. Here we will give some related results as consequences of Theorem 2.13.

Corollary 2.14.

It is provided that denominators are not equal to zero.

Corollary 2.15.

It is provided that denominators are not equal to zero.

Corollary 2.16.

It is provided that denominators are not equal to zero.

Corollary 2.17.

Proof.

Remark 2.18.

Remark 2.19.

In the case of Riemann-Liouville fractional integral of order , we well use the notation instead of and we replace with with , and with .

Remark 2.20.

In the case of Caputo fractional derivative of order , we well use the notation instead of and we replace with with , and with .

Remark 2.21.

In the case of fractional derivative, we will use the notation instead of and we replace with with , and with .

## 3. Exponential Convexity

Lemma 3.1.

Then is strictly convex on for each .

Proof.

Since for all , , therefore, is strictly convex on for each .

Theorem 3.2.

Then the following statements are valid.

(b)The function is exponentially convex on .

- (c)

which is equivalent to (3.4).

Corollary 3.3.

Then the statement of Theorem 3.2 with instead of is valid.

Corollary 3.4.

Then the statement of Theorem 3.2 with instead of is valid.

Corollary 3.5.

Then the statement of Theorem 3.2 with instead of is valid.

In the following theorem, we prove the monotonicity property of defined in (2.52).

Theorem 3.6.

Proof.

which is equivalent to (3.14) for , .

For , , we get the required result by taking limit in (3.16).

Corollary 3.7.

Corollary 3.8.

Corollary 3.9.

## Authors’ Affiliations

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## Copyright

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