On the maximal term of series in systems of functions in a disk

Abstract

For an entire transcendental function $f$ and for a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty$, a series $A(z)=\sum\limits_{n=1}^{\infty}a_nf(\lambda_n z)$ is said to be regularly convergent in $\{z: |z|<R[A]<+\infty\}$, if $\mathfrak{M}(r,A)=\sum\limits_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in [0, R[A])$, where $R[A]$ is the radius of a regular convergence of $A(z)$ and $M_f(r)=\max\{|f(z)|:|z|=r\}$.

We have found conditions on $(\lambda_n)$ and $f$, under which $\ln\mathfrak{M}(r,A)\sim \ln\,\mu(r,A)$ as $r\to R[A]$, where $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n):n\ge 1\}$ is the maximal term of the series.

At the end of the paper, an unresolved problem is stated.