Paley's inequality of integral transform type

Abstract

Let $\{n_k\}_{k=1}^\infty$ be a sequence of positive integers with Hadamard gap. For an analytic function $F(z) = \sum_{n=0}^{\infty}a_n z^n$ in the unit disc satisfying $\sup_{0 \lt r \lt 1}$ $\int_0^{2\pi} |F(re^{i\theta})|\, d\theta \lt \infty$, the inequality $( \sum_{k=1}^{\infty}|a_{n_k}|^2 )^{1/2} \lt \infty$ holds, which is familiar as Paley's inequality. In this paper, an integral transform version of this inequality is established.