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# On an Inequality of H. G. Hardy

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 264347 (2010)

## Abstract

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.

## 1. Introduction

First, let us recall some facts about fractional derivatives needed in the sequel, for more details see, for example, [1, 2].

Let . By , we denote the space of all functions on which have continuous derivatives up to order , and is the space of all absolutely continuous functions on . By , we denote the space of all functions with . For any , we denote by the integral part of (the integer satisfying ), and is the ceiling of (). By , we denote the space of all functions integrable on the interval , and by the set of all functions measurable and essentially bounded on . Clearly, .

We start with the definition of the *Riemann-Liouville fractional integrals*, see [3]. Let , be a finite interval on the real axis . The Riemann-Liouville fractional integrals and of order are defined by

respectively. Here is the Gamma function. These integrals are called the left-sided and the right-sided fractional integrals. We denote some properties of the operators and of order , see also [4]. The first result yields that the fractional integral operators and are bounded in , , that is

where

Inequality (1.3), that is the result involving the left-sided fractional integral, was proved by H. G. Hardy in one of his first papers, see [5]. He did not write down the constant, but the calculation of the constant was hidden inside his proof.

Throughout this paper, all measures are assumed to be positive, all functions are assumed to be positive and measurable, and expressions of the form , , and are taken to be equal to zero. Moreover, by a weight , we mean a nonnegative measurable function on the actual interval or more general set.

The paper is organized in the following way. After this Introduction, in Section 2 we state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy since 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. We conclude this paper with new results involving mixed Riemann-Liouville fractional integrals.

## 2. The Main Results

Let and be measure spaces with positive -finite measures, and let be a nonnegative function, and

Throughout this paper, we suppose that a.e. on , and by a weight function (shortly: a weight), we mean a nonnegative measurable function on the actual set. Let denote the class of functions with the representation

where is a measurable function.

Our first result is given in the following theorem.

Theorem 2.1.

Let be a weight function on , a nonnegative measurable function on , and be defined on by (2.1). Assume that the function is integrable on for each fixed . Define on by

If is convex and increasing function, then the inequality

holds for all measurable functions and for all functions .

Proof.

By using Jensen's inequality and the Fubini theorem, since is increasing function, we find that

and the proof is complete.

As a special case of Theorem 2.1, we get the following result.

Corollary 2.2.

Let be a weight function on and . denotes the Riemann-Liouville fractional integral of . Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that and , so (2.7) follows.

Remark 2.3.

In particular for the weight function in Corollary 2.2, we obtain the inequality

Although (2.4) holds for all convex and increasing functions, some choices of are of particular interest. Namely, we will consider power function. Let and the function be defined by , then (2.9) reduces to

Since and , then we obtain that the left hand side of (2.10) is

and the right-hand side of (2.10) is

Combining (2.11) and (2.12), we get

Taking power on both sides, we obtain (1.3).

Corollary 2.4.

Let be a weight function on and . denotes the Riemann-Liouville fractional integral of . Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Similar to the proof of Corollary 2.2.

Remark 2.5.

In particular for the weight function in Corollary 2.4, we obtain the inequality

Let and the function be defined by , then (2.16) reduces to

Since and , then we obtain that the left hand side of (2.17) is

and the right-hand side of (2.17) is

Combining (2.18) and (2.19), we get

Taking power on both sides, we obtain (1.3).

Theorem 2.6.

Let ,, , and denote the Riemann-Liouville fractional integral of , then the following inequalities

hold, where .

Proof.

We will prove only inequality (2.21), since the proof of (2.22) is analogous. We have

Then by the Hölder inequality, the right-hand side of the above inequality is

Thus, we have

Consequently, we find

and we obtain

Remark 2.7.

For , inequalities (2.21) and (2.22) are refinements of (1.3) since

We proved that Theorem 2.6 is a refinement of (1.3), and Corollaries 2.2 and 2.4 are generalizations of (1.3).

Next, we give results with respect to the *generalized Riemann-Liouville fractional derivative*. Let us recall the definition, for details see [1, page 448].

We define the generalized Riemann-Liouville fractional derivative of of order by

where .

For , we say that has an fractional derivative in , if and only if

(1), ,

(2),

(3).

Next, lemma is very useful in the upcoming corollary (see [1, page 449] and [2]).

Lemma 2.8.

Let and let have an fractional derivative in and let

then

for all .

Corollary 2.9.

Let be a weight function on , and let assumptions in Lemma 2.8 be satisfied. Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that . Replace by . Then, by Lemma 2.8, and we get (2.33).

Remark 2.10.

In particular for the weight function , in Corollary 2.9, we obtain the inequality

Let and the function be defined by , then after some calculation, we obtain

Next, we define *Canavati-type fractional derivative**-fractional derivative of*, for details see [1, page 446]. We consider

. Let . We define the generalized -fractional derivative of over as

the derivative with respect to .

Lemma 2.11.

Let , where and . Assume that , , then

for all .

Corollary 2.12.

Let be a weight function on , and let assumptions in Lemma 2.11 be satisfied. Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Similar to the proof of Corollary 2.9.

Remark 2.13.

In particular for the weight function , in Corollary 2.12, we obtain the inequality

Let and the function be defined by , then (2.42) reduces to

Since and , then we obtain

Taking power on both sides of (2.44), we obtain

When , we find that

that is,

In the next corollary, we give results with respect to the *Caputo fractional derivative*. Let us recall the definition, for details see [1, page 449].

Let , , . The Caputo fractional derivative is given by

for all . The above function exists almost everywhere for .

Corollary 2.14.

Let be a weight function on and . denotes the Caputo fractional derivative of . Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that . Replace by , so becomes and (2.50) follows.

Remark 2.15.

In particular for the weight function , in Corollary 2.14, we obtain the inequality

Let and the function be defined by , then after some calculation, we obtain

Taking power on both sides, we obtain

Theorem 2.16.

Let , , , denotes the Caputo fractional derivative of , then the following inequality

holds.

Proof.

Similar to the proof of Theorem 2.6.

The following result is given [1, page 450].

Lemma 2.17.

Let , , and . Assume that such that , , and , then and

for all .

Corollary 2.18.

Let be a weight function on and . denotes the Caputo fractional derivative of , and assumptions in Lemma 2.17 are satisfied. Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that . Replace by , so becomes and (2.58) follows.

Remark 2.19.

In particular for the weight function , in Corollary 2.18, we obtain the inequality

Let and the function be defined by , then after some calculation, we obtain

For , we obtain

We continue with definitions and some properties of the *fractional integrals of a function**with respect to given function*. For details see, for example, [3, page 99].

Let , be a finite or infinite interval of the real line and . Also let be an increasing function on and a continuous function on . The left- and right-sided fractional integrals of a function with respect to another function in are given by

respectively.

Corollary 2.20.

Let be a weight function on , and let be an increasing function on , such that is a continuous function on and . denotes the left-sided fractional integral of a function with respect to another function in . Define on by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that , so (2.66) follows.

Remark 2.21.

In particular for the weight function , in Corollary 2.20, we obtain the inequality

Let and the function be defined by , then (2.68) reduces to

Since and , is increasing, then and and we obtain

Remark 2.22.

If , then reduces to Riemann-Liouville fractional integral and (2.70) becomes (2.13).

Analogous to Corollary 2.20, we obtain the following result.

Corollary 2.23.

Let be a weight function on , and let be an increasing function on , such that is a continuous function on and . denotes the right-sided fractional integral of a function with respect to another function in . Define on by

If is convex and increasing function, then the inequality

holds.

Remark 2.24.

In particular for the weight function , and for function ,, we obtain after some calculation

Remark 2.25.

If , then reduces to Riemann-Liouville fractional integral and (2.73) becomes (2.20).

The refinements of (2.70) and (2.73) for are given in the following theorem.

Theorem 2.26.

Let , , , and denote the left-sided and right-sided fractional integral of a function with respect to another function in , then the following inequalities:

hold.

We continue by defining *Hadamard type fractional integrals*.

Let be a finite or infinite interval of the half-axis and . The left- and right-sided Hadamard fractional integrals of order are given by

respectively.

Notice that Hadamard fractional integrals of order are special case of the left- and right-sided fractional integrals of a function with respect to another function in , where , so (2.70) reduces to

and (2.73) becomes

Also, from Theorem 2.26 we obtain refinements of (2.77) and (2.78), for ,

Some results involving Hadamard type fractional integrals are given in [3, page 110]. Here, we mention the following result that can not be compared with our result.

Let , , and , then the operators and are bounded in as follows:

where

Now we present the definitions and some properties of the *Erdélyi-Kober type fractional integrals*. Some of these definitions and results were presented by Samko et al. in [4].

Let , be a finite or infinite interval of the half-axis . Also let , , and . We consider the left- and right-sided integrals of order defined by

respectively. Integrals (2.82) and (2.83) are called the Erdélyi-Kober type fractional integrals.

Corollary 2.27.

Let be a weight function on , denotes the hypergeometric function, and denotes the Erdélyi-Kober type fractional left-sided integral. Define by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that , so (2.85) follows.

Remark 2.28.

In particular for the weight function where in Corollary 2.27, we obtain the inequality

where .

Corollary 2.29.

Let be a weight function on , denotes the hypergeometric function, and denotes the Erdélyi-Kober type fractional right-sided integral. Define by

If is convex and increasing function, then the inequality

holds.

Proof.

Applying Theorem 2.1 with , , ,

we get that , so (2.89) follows.

Remark 2.30.

In particular for the weight function where in Corollary 2.29, we obtain the inequality

where .

In the next corollary, we give some results related to the *Caputo radial fractional derivative*. Let us recall the following definition, see [1, page 463].

Let , , , such that , for all , where for and . We call the Caputo radial fractional derivative as the following function:

where , that is, , , .

Clearly,

Corollary 2.31.

Let be a weight function on , and denotes the Caputo radial fractional derivative of . Define on by

If is convex and increasing, then the inequality

holds.

Proof.

Apply Theorem 2.1 with , , , and

Then replace by , so (2.95) follows.

Remark 2.32.

In particular for the weight function , , we obtain the following inequality:

Let and be defined by , then (2.97) becomes

Since and , we obtain

Taking power on both sides, we get

If , then

If , then

Now, we continue with the *Riemann-Liouville radial fractional derivative of**of order*, but first we need to define the following: let stand for the Borel class on space and define the measure on by

Now, let .

For a fixed , we define

where

The above led to the following definition of Riemann-Liouville radial fractional derivative. For details see [1, page 466]. Let ,, , and is the spherical shell. We define

where

If , define

We call the Riemann-Liouville radial fractional derivative of of order .

The following result is given in [1, page 466].

Lemma 2.33.

Let , , , with . Assume that , for every , and that is measurable on for every . Also assume that there exists for every and for every , and is measurable on . Suppose that there exists ,

We suppose that , , for every , then

is valid for every , that is, true for every and for every , .

Corollary 2.34.

Let be a weight function on . Let the assumption of the Lemma 2.33 be satisfied, and denotes the Riemann-Liouville radial fractional derivative of . Define on by

If is convex and increasing, then the inequality

holds.

Proof.

Applying Theorem 2.1 with ,

we get that . Replace by , and then from the above Lemma 2.33, we get . This will give us (2.112).

Remark 2.35.

In particular for the weight function , in above Corollary 2.34 and for , we obtain, after some calculation, the following inequality:

If , then

In the previous corollaries, we derived only inequalities over some subsets of . However, Theorem 2.1 covers much more general situations. We conclude this paper with multidimensional fractional integrals. Such operations of fractional integration in the - dimensional Euclidean space , () are natural generalizations of the corresponding one-dimensional fractional integrals and fractional derivatives, being taken with respect to one or several variables.

For and , we use the following notations:

and by , we mean .

The *partial Riemann-Liouville fractional integrals* of order with respect to the th variable are defined by

respectively. These definitions are valid for functions defined for and , respectively.

Next, we define the *mixed Riemann-Liouville fractional integrals* of order as

Corollary 2.36.

Let be a weight function on and . denotes the mixed partial Riemann-Liouville fractional integral of . Define on by

If is convex and increasing function, then the inequality

holds for all measurable functions .

Proof.

Applying Theorem 2.1 with ,

we get that and , so (2.121) follows.

Corollary 2.37.

Let be a weight function on and . denotes the mixed partial Riemann-Liouville fractional integral of . Define on by

If is convex and increasing function, then the inequality

holds for all measurable functions .

Remark 2.38.

Analogous to Remarks 2.3 and 2.5, we obtain multidimensional version of inequality (1.3) for as follows:

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## Acknowledgments

The authors would like to express their gratitude to professor S. G. Samko for his help and suggestions and also to the careful referee whose valuable advice improved the final version of this paper. The research of the second and third authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant no. 117-1170889-0888.

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Iqbal, S., Krulić, K. & Pečarić, J. On an Inequality of H. G. Hardy.
*J Inequal Appl* **2010**, 264347 (2010). https://doi.org/10.1155/2010/264347

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DOI: https://doi.org/10.1155/2010/264347