On the finiteness of the $l$-$\mathfrak{M}$-index of entire functions represented by series in а system of functions

Abstract

UDC517.5

Let $f$ be an entire transcendental function and let $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty.$ Suppose that the series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ is regularly convergent in ${\mathbb C},$ i.e., $\mathfrak{M}(r,A):=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in [0,+\infty).$  For a positive function $l$ continuous on $[0,\,+\infty),$ the function $A$ is called a function of bounded $l$-$\mathfrak{M}$-index if there exists $N\in{\Bbb Z}_+$ such that $\dfrac{\mathfrak{M}(r,A^{(n)})}{n!l^n(r)}\le\max\left\{\dfrac{\mathfrak{M}(r,A^{(k)})}{k!l^k(r)}\colon 0\le k\le N\right\}$ for all $n\in{\Bbb Z}_+$ and all $r\in [0,+\infty).$  We study the properties of growth of the functions of bounded $l$-$\mathfrak{M}$-index  and formulate some unsolved problems.