A New Hilbert-Type Integral Inequality with the Homogeneous Kernel of Real Degree Form and the Integral in Whole Plane

Zi Tian Xie,K Raja Rama Gandhi,Zeng Zheng

doi:10.18052/www.scipress.com/bmsa.3.71

Abstract

In this paper, we build a new Hilbert's inequality with the homogeneous kernel of real order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using ψ function.

Highlights

(1.1) where the constant factor π is the best possible
Inequality (1.1) is well-known as Hilbert's integral inequality,which has been extended by Hardy-Riesz as [2]
Recent XIE Zitin and ZHOU Qinghua prove that the expression of the -function admitsa finite expression in elementary function for rational number z,and prove that [6]

Summary

INTRODUCTION

Inequality (1.1) is well-known as Hilbert's integral inequality,which has been extended by Hardy-Riesz as [2]. We have the following Hardy-Hilbert's integral inequality:. Hilbert's inequality attracts some attention in recent years.inequalities (1.1)and(1.2) have many generalizations and variations. (1.1) has been strengthened by Yang and others( including double series inequalities ). In 2008, Zitian Xie and Zheng Zeng gave a new Hilbert-type Inequality [4] as follows :. Where the constant factor is the best possible.

72 Assume that

SOME LEMMAS

MAIN RESULTS

SOME EXAMPLES