Geodesic
Power series with fast decreasing coefficients
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 38, Tome 376 (2010), pp. 167-175 
A. M. Chirikov. Power series with fast decreasing coefficients. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 38, Tome 376 (2010), pp. 167-175. http://geodesic.mathdoc.fr/item/ZNSL_2010_376_a6/
@article{ZNSL_2010_376_a6,
     author = {A. M. Chirikov},
     title = {Power series with fast decreasing coefficients},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--175},
     year = {2010},
     volume = {376},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_376_a6/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] N. A. Shirokov, Analytic functions smooth up to the boundary, Lect. Notes Math., 1312, Springer-Verlag, 1988 | MR | Zbl

[2] M. V. Fedoryuk, Metod perevala, Moskva, 2010

[3] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, GITTL, Moskva–Leningrad, 1950

[4] A. I. Markushevich, Teoriya analiticheskikh funktsii, v. 2, izd. 2, Nauka, Moskva, 1968 | Zbl

[5] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnoi peremennoi, Nauka, Moskva, 1965 | Zbl