Minimum modulus of lacunary power series and $h$-measure of exceptional sets

Abstract

We consider some generalizations of Fenton theorem for the entire functions represented by lacunary power series. Let $f(z)=\sum_{k=0}^{+\infty}f_kz^{n_k}$, where $(n_k)$ is a strictly increasing sequence of non-negative integers. We denote by \begin{align*} &M_f(r)=\max\{|f(z)|\colon |z|=r\}, \\ &m_f(r)=\min\{|f(z)|\colon |z|=r\}, \\ & \mu_f(r)=\max\{|f_k|r^{n_k}\colon k\geq 0\} \end{align*} the maximum modulus, the minimum modulus and the maximum term of $f,$ respectively. Let $h(r)$ be a positive continuous function increasing to infinity on $[1,+\infty)$ with a non-decreasing derivative. For a measurable set $E\subset [1,+\infty)$ we introduce $h-\mathrm{meas}\,(E)=\int_{E}\frac{dh(r)}{r}.$ In this paper we establish conditions guaranteeing that the relations $$ M_f(r)=(1+o(1)) m_f(r),\quad M_f(r)=(1+o(1))\mu_f(r) $$ are true as $r\to+\infty$ outside some exceptional set $E$ such that $h-\mathrm{meas}\,(E)<+\infty$. For some subclasses we obtain necessary and sufficient conditions. We also provide similar results for entire Dirichlet series.