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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 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
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 ofof 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