On a Multiple Hilbert's Inequality with Parameters

Qiliang Huang

doi:10.1155/2010/309319

Abstract

By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

Highlights

In 1908, Weyl published the following famous Hilbert’s inequality cf. 1
We prove that the constant factor in the new inequality is the best possible and consider its equivalent form
Since 1/pn 1/qn 1, by 2.1 and Holder’s inequality cf. 5, we find that mn−1 1 m1 1 n−1 i1 ami i

Summary

Introduction

In 1908, Weyl published the following famous Hilbert’s inequality cf. 1. Α1−n Γλ n i1 λ ri mipi 1−λα/ri −1 ami i pi The constant factors in the above five inequalities are all the best possible. We prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

Results

Conclusion