On an Inequality of H. G. Hardy
© Sajid Iqbal et al. 2010
Received: 18 June 2010
Accepted: 16 October 2010
Published: 24 October 2010
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.
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, .
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 . 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
Our first result is given in the following theorem.
and the proof is complete.
As a special case of Theorem 2.1, we get the following result.
Similar to the proof of Corollary 2.2.
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].
Similar to the proof of Corollary 2.9.
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].
Similar to the proof of Theorem 2.6.
The following result is given [1, page 450].
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].
Analogous to Corollary 2.20, we obtain the following result.
We continue by defining Hadamard type fractional integrals.
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 .
respectively. Integrals (2.82) and (2.83) are called the Erdélyi-Kober type fractional integrals.
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].
The following result is given in [1, page 466].
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.
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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