TY - JOUR
AU - Mulyava, O.M.
AU - Sheremeta, M.M.
PY - 2013/12/30
Y2 - 2020/11/29
TI - Remarks on sufficient conditions of belonging of analytic functions to convergence classes
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 5
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.5.2.298-304
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1320
SP - 298-304
AB - It is well known that if Taylor's coefficients $f_n$ of an entire functions $f$ satisfy the conditions $|f_k|/|f_{k+1}|
earrow +\infty$ 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 $|f_k|/|f_{k+1}|
earrow +\infty$ one can replace on the condition $l_{k-1}l_{k+1}l^{-2}_{k}|f_k|/|f_{k+1}|
earrow +\infty$, 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.
ER -