On Hilbert's Integral Inequality

Abstract

In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff,g∈L2[0,∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π2is also the best value (cf. [1, Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x)=2/π∫∞0(c(t2x)/(1+t2))dt−c(x),1−c(x)+c(y)≥0, andf,g≥0 (cf. [2]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [2].