Lacunary series in weighted spaces of analytic functions

Abstract

We improve a recent result of Yang and Xu (Arch. Math. 96 (2011), 151–160) by proving that if ψ is a normal function on [1, ∞) and ${f(z)=\sum_{n=0}^\infty a_n z^{k_n}}$ (|z| < 1) is an analytic function with Hadamard gaps, then

$$\frac 1C \sup_{n\ge 0} \frac{|a_n|}{\psi(k_n)} \le \sup_{0 < r < 1} \frac{|f(r\zeta)|}{\psi(1/(1-r))} \le C\sup_{n\ge 0} \frac{|a_n|}{\psi(k_n)}, \quad |\zeta|=1,$$

where C is a constant independent of ζ and {a _n }.