A Hardy–Hilbert's type inequality with gamma function and its applications

Abstract

In this paper, it is shown that a Hardy–Hilbert integral type inequality related to gamma function can be established by introducing a proper weight function of the form (Γ (x))1−λ (Γ′(x))1−r (x≥2, r>1, 1−(q/p)<λ≤2, p≥q>1). And the double series analogue of the Hardy–Hilbert type inequality is also built. In particular, for case p=2, some new extensions of the classical Hilbert's inequality (including integral and discrete forms) are obtained. As applications, some extensions on Hardy–Litlewood's theorem are given.