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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2010 Volume 376, Pages 167–175 (Mi znsl3622)

Power series with fast decreasing coefficients

A. M. Chirikov

Herzen State Pedagogical University of Russia, St. Petersburg, Russia

Abstract: Let $f(x)=\sum_{n=0}^\infty a_nx^n$ be an analytic function in the unit disc such that for some $\lambda>1$, $C_0,C_1,C_2,C_3>0$ we have
$$ |f(x)|\le C_0\exp(-C_1|\log(1-x)|^\lambda),\qquad\frac12<x<1 $$
and
$$|a_n|\le C_2\exp\biggl(-C_3\frac{\sqrt n}{\log(n+2)}\biggr),\qquad n\ge0. $$
Then $f\equiv0$. Bibl. – 5 titles.

Key words and phrases: Taylor coefficients, power series, decreasing on a radius, uniqueness theorems for analytic functions.

UDC: 517.537.3

Received: 12.05.2010


 English version:
Journal of Mathematical Sciences (New York), 2011, 172:2, 270–275

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© Steklov Math. Inst. of RAS, 2024