A New Extension of Hardy-Hilbert’s Inequality in the Whole Plane

Yang, Bicheng; Chen, Qiang

doi:https://doi.org/10.1155/2016/9197476

Journal of Function Spaces

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Research Article | Open Access

Volume 2016 | Article ID 9197476 | https://doi.org/10.1155/2016/9197476

A New Extension of Hardy-Hilbert’s Inequality in the Whole Plane

Bicheng Yang¹and Qiang Chen²

Academic Editor: Raúl E. Curto

Received27 Mar 2016

Accepted30 Jun 2016

Published08 Sept 2016

Abstract

By the use of weight coefficients and Hermite-Hadamard’s inequality, a new extension of Hardy-Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.

1. Introduction

Suppose that , , , , , , and We have the following well known Hardy-Hilbert’s inequality:where the constant factor is the best possible one (cf. [1]). The more accurate form of (1) was given as follows (cf. [2], Theorem ):where the constant factor is still the best possible one. For , inequality (2) reduces to (1).

In 2011, Yang gave an extension of (2) as follows (cf. [3]): If , , , , , and , thenwhere the constant factor is the best possible one and is the beta function defined by (cf. [4])For , , and , (3) reduces to (2). Some other results related to (1)–(3) are provided by [5–22].

In this paper, by the use of weight coefficients and Hermite-Hadamard’s inequality, an extension of (3) in the whole plane is given as follows: For , , , and , we haveMoreover, a generation of (5) with multiparameters and a best possible constant factor is proved. The equivalent forms, the operator expressions, and a few particular inequalities are also considered.

2. Some Lemmas

First, we make appointment that , , , , , , , and, for , In particular, for , we indicate

Definition 1. Define the following weight coefficients:where

Lemma 2. If , then, for , one haswhere

Proof. For , we setand then, for ,We find It is evident that, for fixed , , both and are strictly decreasing and strictly convex with respect to , satisfyingBy Hermite-Hadamard’s inequality (cf. [23]), we findSetting in the above first (second) integral, by simplification, we findBy (14), since both and are strictly decreasing, we still haveWe obtainand then we have (10) and (11).

In the same way, we still have the following.

Lemma 3. If , then, for , one haswhere

Lemma 4. If and , then, for ,

Proof. We haveFor , we findHence, we have (22).

3. Main Results and Operation Expressions

Theorem 5. If , ,then one has the following equivalent inequalities:

Proof. By Hölder’s inequality (cf. [23]) and (9), we haveBy (20), we haveBy (10), we have (27).
By Hölder’s inequality (cf. [23]), we have Then by (27), we have (26).
On the other hand, assuming that (26) is valid, we setThen it follows that By (29), we find If , then (27) is evidently valid; if , then, by (26), we have namely, (27) follows, which is equivalent to (26).

Theorem 6. As regards the assumptions of Theorem 5, the constant factor in (26) and (27) is the best possible one.

Proof. For any , we set , , and Then by (22) and (10), we findIf there exists a constant , such that (26) is valid when replacing by , then, in particular, we have ; namely,It follows thatnamely, Hence, is the best possible constant factor of (26). The constant factor in (27) is still the best possible one. Otherwise, we would reach a contradiction by (30) that the constant factor in (26) is not the best possible one.

We set functions and as follows:wherefrom We also set the following weight normed spaces:

Then, for , , and , in view of (27), we have ; namely,

Definition 7. Define a Hilbert-type operator as follows: for any , there exists a unique representation One also defines the formal inner product of and as follows:Then for , we may rewrite (26) and (27) as follows:We define the norm of operator as follows:Since, by Theorem 6, the constant factor in (42) is the best possible one, we have

Remark 8. (i) For , (26) reduces toHence, (26) is a more accurate inequality of (45). In particular, for in (45), we have the following new inequality: (ii) For in (26), we have (5). For and , (5) reduces toIn particular, for , we haveIf , then (48) reduces towhich is an extension of (1).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by Science and Technology Planning Project of Guangdong Province (no. 2013A011403002) and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (no. 2015ARF25). The authors are grateful for their help.

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Copyright

Copyright © 2016 Bicheng Yang and Qiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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