On a new Hardy-Littlewood-Polya’s inequality with multi-parameters and its applications

(1.1) and (1.2) is the well known Hardy-Littlewood-Polya’s inequality. In connection with applications in analysis, their generalizations and variants have received considerable interest recent years. Firstly, by means of introducing a parameter, two forms of extended Hardy-Littlewood-Polya’s inequality are obtained by Hu in [2] as follows. (1) Let λ > 0, p > 1, 1 p+ 1 q=1, f(x), g(y) ≥ 0, F (x) = x (1−λ)/qf(x) ∈ L(0,∞), G(y) = y(1−λ)/pg(y) ∈ L(0,∞), K(x, y) ≥ 0,K(x, y) is homogeneous -1 expression, and ∫∞ 0 K(ω, 1)ωλ/p−1dω = k, then ∫ ∞