Remarks on sufficient conditions of belonging of analytic functions to convergence classes
Keywords:
entire function, analytic function in a disk, convergence class
Published online:
2013-12-30
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.
How to Cite
(1)
Mulyava O., Sheremeta M. Remarks on Sufficient Conditions of Belonging of Analytic Functions to Convergence Classes. Carpathian Math. Publ. 2013, 5 (2), 298-304.