Analogues of the Lockhart-Straus inequality and two- member asymptotics of series in systems of functions

Abstract

For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty$ let $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ be a series in the system ${f(\lambda_nz)}$ regularly convergent in $\{z:|z|<R[A]\}$ that is $\mathfrak{M}(r,A)=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for $r\in [0, R[A])$, where
$M_f(r)=\max\{|f(z)|:\,|z|=r\}$ for $r\in [0, +\infty)$, and let $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n):\,n\ge 1\}$ be the maximal term.

Estimates of $\mathfrak{M}(r,A)$ by $\mu(r,A)$ are obtained, which are analogues of the well-known Lockhart-Straus inequality (1985). The application of the obtained results is indicated to study the relationship between the growth of $\mathfrak{M}(r,A)$ and $\mu(r,A)$ in terms of two-member asymptotics.