Remarks on sufficient conditions of belonging of analytic functions to convergence classes

Abstract

It is well known that if Taylor's coefficients  $f_n$ of an entire functions $f$ satisfy the conditions $|f_k|/|f_{k+1}|\nearrow +\infty$ as $k\to\infty$ and $\sum\limits_{k=1}^{\infty}|f_k|^{\varrho/k}<+\infty$ then $f$ belongs to Valiron convergence class. It is proved that in the statement the condition $|f_k|/|f_{k+1}|\nearrow +\infty$ one can replace on the condition $l_{k-1}l_{k+1}l^{-2}_{k}|f_k|/|f_{k+1}|\nearrow +\infty$, where $(l_k)$ is a positive sequence such that $\root{k}\of{l_k/l_{k+1}}\asymp 1$ as $k\to\infty$. Analogous problems are solved for another convergence classes of entire and analytic functions in the unit disk.