Densities in Fabry’s theorem. (English) Zbl 1191.30001

Eremenko, Alexandre

Ill. J. Math. 52, No. 4, 1277-1290 (2008).

A classical result of A. Pringsheim [Math. Ann. XLIV. 41–56 (1894; JFM 25.0389.01)] states that, for every power series \(\sum^\infty_{m=0} a_mz^m\) with nonnegative coefficients and \[ \limsup_{m\to\infty}|a_m|^{1/m}= 1, \] the point \(1\) is singular. This was generalized by E. Fabry [Acta Math. 22, 65–88 (1898; JFM 29.0209.04)] to guarantee the existence of a singular point on a closed arc of \(\{|z|=1\}\) centred at \(1\), subject to certain conditions on the coefficients including a maximum density condition.
Improving a recent improvement by N. U. Arakelyan and V. A. Martirosyan [Izv. Akad. Nauk Arm. SSR, Mat. 22, No. 1, 3–21 (1987; Zbl 0626.30002); Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 22, No. 1, 1–19 (1987; Zbl 0639.30004)], the author further generalizes Pringsheim’s result replacing maximum density by an interior density of Beurling-Malliavin type.

Reviewer: D. A. Brannan (Milton Keynes)

MSC:

30B10	Power series (including lacunary series) in one complex variable
30B40	Analytic continuation of functions of one complex variable

Keywords:

power series; Fabry’s theorem; Pringsheim’s theorem; Beurling-Malliavin density

Citations:

Zbl 0626.30002; Zbl 0639.30004; JFM 25.0389.01; JFM 29.0209.04

References:

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