On an Inequality of H. G. Hardy

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

(1.1)

(1.2)

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

(1.3)

where

(1.4)

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

(2.1)

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

(2.2)

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

(2.3)

If is convex and increasing function, then the inequality

(2.4)

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

(2.5)

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

(2.6)

If is convex and increasing function, then the inequality

(2.7)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

and the right-hand side of (2.10) is

(2.12)

Combining (2.11) and (2.12), we get

(2.13)

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

(2.14)

If is convex and increasing function, then the inequality

(2.15)

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

(2.16)

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

(2.17)

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

(2.18)

and the right-hand side of (2.17) is

(2.19)

Combining (2.18) and (2.19), we get

(2.20)

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

(2.21)

(2.22)

hold, where .

Proof.

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

(2.23)

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

(2.24)

Thus, we have

(2.25)

Consequently, we find

(2.26)

and we obtain

(2.27)

Remark 2.7.

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

(2.28)

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

(2.29)

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

(2.30)

then

(2.31)

for all .

Corollary 2.9.

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

(2.32)

If is convex and increasing function, then the inequality

(2.33)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.34)

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

(2.35)

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

(2.36)

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

(2.37)

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

(2.38)

the derivative with respect to .

Lemma 2.11.

Let , where and . Assume that , , then

(2.39)

for all .

Corollary 2.12.

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

(2.40)

If is convex and increasing function, then the inequality

(2.41)

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

(2.42)

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

(2.43)

Since and , then we obtain

(2.44)

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

(2.45)

When , we find that

(2.46)

that is,

(2.47)

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

(2.48)

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

(2.49)

If is convex and increasing function, then the inequality

(2.50)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.51)

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

(2.52)

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

(2.53)

Taking power on both sides, we obtain

(2.54)

Theorem 2.16.

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

(2.55)

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

(2.56)

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

(2.57)

If is convex and increasing function, then the inequality

(2.58)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.59)

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

(2.60)

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

(2.61)

For , we obtain

(2.62)

We continue with definitions and some properties of the fractional integrals of a functionwith 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

(2.63)

(2.64)

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

(2.65)

If is convex and increasing function, then the inequality

(2.66)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.67)

we get that , so (2.66) follows.

Remark 2.21.

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

(2.68)

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

(2.69)

Since and , is increasing, then and and we obtain

(2.70)

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

(2.71)

If is convex and increasing function, then the inequality

(2.72)

holds.

Remark 2.24.

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

(2.73)

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:

(2.74)

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

(2.75)

(2.76)

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

(2.77)

and (2.73) becomes

(2.78)

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

(2.79)

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:

(2.80)

where

(2.81)

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

(2.82)

(2.83)

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

(2.84)

If is convex and increasing function, then the inequality

(2.85)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.86)

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

(2.87)

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

(2.88)

If is convex and increasing function, then the inequality

(2.89)

holds.

Proof.

Applying Theorem 2.1 with , , ,

(2.90)

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

(2.91)

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:

(2.92)

where , that is, , , .

Clearly,

(2.93)

Corollary 2.31.

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

(2.94)

If is convex and increasing, then the inequality

(2.95)

holds.

Proof.

Apply Theorem 2.1 with , , , and

(2.96)

Then replace by , so (2.95) follows.

Remark 2.32.

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

(2.97)

Let and be defined by , then (2.97) becomes

(2.98)

Since and , we obtain

(2.99)

Taking power on both sides, we get

(2.100)

If , then

(2.101)

If , then

(2.102)

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

(2.103)

Now, let .

For a fixed , we define

(2.104)

where

(2.105)

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

(2.106)

where

(2.107)

If , define

(2.108)

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 ,

(2.109)

We suppose that , , for every , then

(2.110)

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

(2.111)

If is convex and increasing, then the inequality

(2.112)

holds.

Proof.

Applying Theorem 2.1 with ,

(2.113)

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:

(2.114)

If , then

(2.115)

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:

(2.116)

and by , we mean .

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

(2.117)

(2.118)

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

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

(2.119)

Corollary 2.36.

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

(2.120)

If is convex and increasing function, then the inequality

(2.121)

holds for all measurable functions .

Proof.

Applying Theorem 2.1 with ,

(2.122)

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

(2.123)

If is convex and increasing function, then the inequality

(2.124)

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:

(2.125)