Discrete Hilbert type inequality with non-homogeneous kernel

Ban Drascic,T.K Pogány

doi:10.2298/aadm0901088d

Abstract

A new version of discrete Hilbert type inequality is given where the kernel function is non-homogeneous. The main mathematical tools are the representation of the Dirichlet series by means of the Laplace integral, and the H?lder inequality with non-conjugated parameters. Numerous special cases are treated and conditional best constants are discussed.

Highlights

AND PRELIMINARIESLet p be the space of all complex sequences x =∞ n=1 with x p := ∞ 1/p|xn|p < +∞
Fundamental contributions have been given to this classical inequality by
After splitting the kernel function into two Dirichlet series, we evaluate these Dirichlet series by the Holder inequality with non–conjugated parameters p, q, min{p, q} > 1, p−1 + q−1 ≥ 1 [13, p. 57]

Summary

INTRODUCTION

The famous Hilbert’s double series theorem, frequently n=1 called a discrete Hilbert inequality too, reads as follows. Hilbert inequalities with non–homogeneous kernels were studied in [2], [3], [7],. As already pointed out in [17], the standard way in deriving Hilbert’s inequality is to apply the Holder inequality to a suitably transformed Hilbert type. Discrete Hilbert type inequality, Dirichlet–series, Holder inequality, Jacobi Theta function, non–homogeneous kernel. K(m, n) ambn m,n=1 where a, b are nonnegative; K(·, ·) we call kernel function of the double series (2). To obtain discrete Hilbert type inequalities (or in other words - double series theorems) one derives sharp upper bounds for HaK,b in terms of weighted p–norms of a, b.

MAIN RESULT

SPECIAL CASES

FINAL REMARKS