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

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.

1. Introduction

Let be a nonnegative kernel. Consider a function , where , and the representation of is

(1.1)

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.

Let and for all . Also let be a function such that is convex and increasing for . Then

(1.2)

where

(1.3)

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

Definition 1.2 (see [2]).

Let be an interval, and let be convex on . Then, for such that , the following inequality holds:

(1.4)

Let us recall the following definition.

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

A function is exponentially convex if it is continuous and

(1.5)

and all choices of ,.

The following proposition is useful to prove the exponential convexity.

Proposition 1.4 (see [4]).

Let . The following statements are equivalent.

(i) is exponentially convex.

(ii) is continuous, and

(1.6)

for every ,, and , .

Corollary 1.5.

If is exponentially convex, then is -convex; that is,

(1.7)

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.

Let , and for all . Also let be an interval, let be convex, and let , . Then

(2.1)

where

(2.2)

Proof.

Since and , we have

(2.3)

By Jensen's inequality, we get

(2.4)

Remark 2.2.

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

Remark 2.3.

Let us note that Theorem 1.1 follows from Theorem 2.1. Indeed, let the condition of Theorem 1.1 be satisfied, and let ; that is,

(2.5)

So, by Theorem 2.1, we have

(2.6)

On the other hand, is increasing function, we have

(2.7)

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

If and , then the Riemann-Liouville fractional integral is defined by

(2.8)

We will use the following kernel in the upcoming corollary:

(2.9)

Corollary 2.4.

Let , and for all . Also let be an interval, let be convex, , , and have Riemann-Liouville fractional integral of order . Then

(2.10)

where

(2.11)

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

Let and . Then the Caputo fractional derivative (see [5, p. 270]) of order α for a function is defined by

(2.12)

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

Here we use the following kernel in the upcoming corollary:

(2.13)

Corollary 2.5.

Let , and for all . Also let be an interval, let be convex, , , and have Caputo fractional derivative of order . Then

(2.14)

where

(2.15)

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.

Let has an fractional derivative in , and

(2.16)

Then

(2.17)

for all .

Clearly

(2.18)

hence

(2.19)

Now we use the following kernel in the upcoming corollary:

(2.20)

Corollary 2.7.

Let , has an fractional derivative in , and for all . Also let for , let be convex, and , . Then

(2.21)

where

(2.22)

Lemma 2.8.

Let , and let be a compact interval, such that

(2.23)

Consider two functions defined as

(2.24)

Then and are convex on .

Proof.

We have

(2.25)

that is are convex on .

Theorem 2.9.

Let , let be a compact interval, , and for all . Also let , , be nonconstant, and let be given in (2.2). Then there exists such that

(2.26)

Proof.

Since and is a compact interval, therefore, suppose that , . Using Theorem 2.1 for the function defined in Lemma 2.8, we have

(2.27)

From Remark 2.2, we have

(2.28)

Therefore, (2.27) can be written as

(2.29)

We have a similar result for the function defined in Lemma 2.8 as follows:

(2.30)

Using (2.29) and (2.30), we have

(2.31)

By Lemma 2.8, there exists such that

(2.32)

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.

Let , let be a compact interval, , and for all . Also let , , let be nonconstant, let be given in (2.11), and , have Riemann-Liouville fractional integral of order . Then there exists such that

(2.33)

Corollary 2.11.

Let , let be compact interval, , and for all . Also let , , let be nonconstant, let be given in (2.15), and have Caputo derivative of order . Then there exists such that

(2.34)

Corollary 2.12.

Let , let be a compact interval, has an fractional derivative, and for all . Let for , , , let be nonconstant, and let be given in (2.22). Then there exists such that

(2.35)

Theorem 2.13.

Let , let be a compact interval, , and for all . Also let be nonconstant, and let be given in (2.2). Then there exists such that

(2.36)

It is provided that denominators are not equal to zero.

Proof.

Let us take a function defined as

(2.37)

where

(2.38)

By Theorem 2.9 with , we have

(2.39)

Since

(2.40)

so we have

(2.41)

This implies that

(2.42)

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.

Let , let be a compact interval, , and for all . Also let , , let be nonconstant, let be given in (2.11), and , have Riemann-Liouville fractional derivative of order . Then there exists such that

(2.43)

It is provided that denominators are not equal to zero.

Corollary 2.15.

Let , let be a compact interval, , and for all . Also let , let be nonconstant, let be given in (2.15), and , have Caputo fractional derivative of order . Then there exists such that

(2.44)

It is provided that denominators are not equal to zero.

Corollary 2.16.

Let , let be a compact interval, has an fractional derivative in , and for all . Also let for , let be nonconstant, and let be given in (2.22). Then there exists such that

(2.45)

It is provided that denominators are not equal to zero.

Corollary 2.17.

Let , let be a compact interval, , and for all . Let , let be nonconstant, and let be given in (2.2). Then, for and , there exists such that

(2.46)

Proof.

We set and , , . By Theorem 2.13, we have

(2.47)

This implies that

(2.48)

This implies that

(2.49)

Remark 2.18.

Since the function is invertible and from (2.46), we have

(2.50)

Now we can suppose that is an invertible function, then from (2.36) we have

(2.51)

We see that the right-hand side of (2.49) is mean, then for distinct it can be written as

(2.52)

as mean in broader sense. Moreover, we can extend these means, so in limiting cases for ,

(2.53)

where and .

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.

Let , and let be a function defined as

(3.1)

Then is strictly convex on for each .

Proof.

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

Theorem 3.2.

Let for all , let be given in (2.2), and

(3.2)

Then the following statements are valid.

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

(3.3)

(b)The function is exponentially convex on .

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

(3.4)

Proof.

• (a) Here we define a new function ,

  

  (3.5)

for , , , where ,

(3.6)

This shows that is convex for . Using Theorem 2.1, we have

(3.7)

From the above result, it shows that the matrix is a positive semidefinite matrix. Specially, we get

(3.8)

• (b) Since

  

  (3.9)

it follows that is continuous for . Then, by using Proposition 1.4, we get the exponential convexity of the function .

• (c) Since is continuous for and using Corollary 1.5, we get that is -convex. Now by Definition 1.2 with and such that , we get

  

  (3.10)

which is equivalent to (3.4).

Corollary 3.3.

Let , and for all . Also let , , have Riemann-Liouville fractional integral of order , let be given in (2.11), and

(3.11)

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

Corollary 3.4.

Let , and for all . Also let , , have Caputo fractional derivative of order , let be given in (2.15), and

(3.12)

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

Corollary 3.5.

Let , has fractional derivative, and for all . Also let for , , , let be given in (2.22), and

(3.13)

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.

Let the assumption of Theorem 3.2 be satisfied, also let be defined in (3.2), and such that . Then the following inequality is true:

(3.14)

Proof.

For a convex function , using the Definition 1.2, we get the following inequality:

(3.15)

with , and . Since by Theorem 3.2 we get that is -convex. We set , , , , , , and . Terefore, we get

(3.16)

which is equivalent to (3.14) for , .

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

Corollary 3.7.

Let , and let the assumption of Corollary 3.3 be satisfied, also let be defined by (3.11). For such that , , then the following inequality holds:

(3.17)

Corollary 3.8.

Let and let the assumption of Corollary 3.4 be satisfied, also let be defined by (3.12). For such that , , then the following inequality holds:

(3.18)

Corollary 3.9.

Let and the assumption of Corollary 3.5 be satisfied, also let be defined by (3.13). For such that , . Then following inequality holds

(3.19)