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

  • O.M. Mulyava National University of Food Technologies, 68 Volodymyrska str., 01601, Kyiv, Ukraine
  • M.M. Sheremeta Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
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.
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How to Cite
(1)
MulyavaO., SheremetaM. Remarks on Sufficient Conditions of Belonging of Analytic Functions to Convergence Classes. Carpathian Math. Publ. 2013, 5 (2), 298-304.