On the logarithmic growth of entire series in a system of functions

Abstract

UDC 517.537

For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty,$ let $A(z)=\displaystyle\sum\nolimits_{n=1}^{\infty}a_nf(\lambda_n z)$ be a series in the system ${f(\lambda_nz)}$ regularly convergent in $\Bbb C,$ i.e., $\mathfrak{M}(r,A)=\displaystyle\sum\nolimits_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for $r\in [0, +\infty),$ where $M_f(r)=\max\{|f(z)|\colon |z|=r\}.$ The logarithmic order $\varrho_{\ln}[A]$ and type $T_{\ln}[A]$ are defined as follows: $\varrho_{\ln}[A]=\limsup_{r\to+\infty}\dfrac{\ln \ln \mathfrak{M}(r,A)}{\ln \ln r}$ and $T_{\ln}[A]=\limsup_{r\to+\infty}\dfrac{\ln \mathfrak{M}(r,A)}{\ln^{\varrho_{\ln}[A]}\,r}.$ Replacing here $\limsup$ with $\liminf,$ we arrive at the definitions of lower logarithmic order and type. We have also deduced the formulas for calculating the indicated characteristics in terms of the coefficients $a_n.$ The relationship between the growth of $\ln \mathfrak{M}(r,A)$ and the decrease in $a_n$ is studied in terms of two-term power asymptotics.