# On an Inequality of H. G. Hardy

- Sajid Iqbal
^{1}Email author, - Kristina Krulić
^{2}and - Josip Pečarić
^{1, 2}

**2010**:264347

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

© Sajid Iqbal et al. 2010

**Received: **18 June 2010

**Accepted: **16 October 2010

**Published: **24 October 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.

## Keywords

## 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, .

*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

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

where is a measurable function.

Our first result is given in the following theorem.

Theorem 2.1.

holds for all measurable functions and for all functions .

Proof.

and the proof is complete.

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

Corollary 2.2.

holds.

Proof.

we get that and , so (2.7) follows.

Remark 2.3.

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

Corollary 2.4.

holds.

Proof.

Similar to the proof of Corollary 2.2.

Remark 2.5.

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

Theorem 2.6.

Proof.

Remark 2.7.

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].

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

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

Lemma 2.8.

Corollary 2.9.

holds.

Proof.

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

Remark 2.10.

*Canavati-type fractional derivative*

*-fractional derivative of*, for details see [1, page 446]. We consider

the derivative with respect to .

Lemma 2.11.

Corollary 2.12.

holds.

Proof.

Similar to the proof of Corollary 2.9.

Remark 2.13.

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].

for all . The above function exists almost everywhere for .

Corollary 2.14.

holds.

Proof.

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

Remark 2.15.

Theorem 2.16.

holds.

Proof.

Similar to the proof of Theorem 2.6.

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

Lemma 2.17.

Corollary 2.18.

holds.

Proof.

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

Remark 2.19.

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].

respectively.

Corollary 2.20.

holds.

Proof.

we get that , so (2.66) follows.

Remark 2.21.

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.

holds.

Remark 2.24.

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.

hold.

We continue by defining *Hadamard type fractional integrals*.

respectively.

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.

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].

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

Corollary 2.27.

holds.

Proof.

we get that , so (2.85) follows.

Remark 2.28.

Corollary 2.29.

holds.

Proof.

we get that , so (2.89) follows.

Remark 2.30.

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].

Corollary 2.31.

holds.

Proof.

Then replace by , so (2.95) follows.

Remark 2.32.

*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

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

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

Lemma 2.33.

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

Corollary 2.34.

holds.

Proof.

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 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.

*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.

Corollary 2.36.

holds for all measurable functions .

Proof.

we get that and , so (2.121) follows.

Corollary 2.37.

holds for all measurable functions .

Remark 2.38.

## Declarations

### 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.

## Authors’ Affiliations

## References

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