On some properties of the maximal term of series in systems of functions

Abstract

For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increa\-sing to $+\infty$ a series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ in the system $\{f(\lambda_nz)\}$ is said to be regularly convergent in ${\mathbb C}$ if $\mathfrak{M}(r,A)=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in (0,+\infty)$, where $ M_f(r)=\max\{|f(z)|\colon |z|=r\}$. Conditions are found on  $(\lambda_n)$ and $f$, under which  $\ln\mathfrak{M}(r,A)\sim \ln \mu(r,A)$ as $r\to+\infty$, where $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n)\colon n\ge 1\}$ is the maximal term of the series. A~formula for finding the lower generalized order $$\lambda_{\alpha,\beta}[A]=\varliminf\limits_{r\to+\infty}\dfrac{\alpha(\ln \mathfrak{M}(r,A))}{\beta(r)}$$ is obtained, where the functions $\alpha$ and $\beta$ are positive, continuous and increasing to $+\infty$.   The open problems are formulated.