Hilbert's double series theorem extended to the case of non-homogeneous kernels

Abstract

Sharp version of celebrated Hilbert's double series theorem is given in the case of non-homogeneous kernel. The main mathematical tools are: the integral representation of Mathieu's (a, λ)-series, the Hölder inequality and an extension of the double series theorem by Yang.

Discrete Hilbert type inequality with non-homogeneous kernel

“…Hardy [4], Mulholland [14], [15], Bonsall [1] and Levin [12]. Discrete Hilbert inequalities with non-homogeneous kernels were studied in [2], [3], [7], [8]- [11], [16], [17], [19]- [24].…”

“…As already pointed out in [17], the standard way in deriving Hilbert's inequality is to apply the Hölder inequality to a suitably transformed Hilbert type double sum expression, that is, to the bilinear form 2H a,b…”

Discrete Multiple Hilbert Type Inequality With Non-Homogeneous Kernel

“…This is the famous discrete Hilbert double series theorem or Hilbert inequality, a topic of interest of many mathematicians now-a-days too. The accustomary approach to deriving Hilbert's inequality is by applying the Hölder inequality to suitably transformed Hilbert type double sum expression, i.e., to the bilinear form ∈ p , n (2−r)/q b r ∈ q , Pogány [7] recently deduced …”

“…secure the finiteness of C λ,ρ [7,Theorem 1], B(·, ·) denotes the Euler Betafunction and λ, ρ : R + → R + are monotone increasing positive functions such that…”

“…Our main goal is to derive a sharp upper bound to (7). Here, and in what follows, I(x) = x denotes the identity function, f −1 (x) stands for the inverse of some f (x) and [X], {X} are the integer part and the fractional part of some X.…”