On Fabry’s gap theorem. (English) Zbl 1090.30003

Puchta, Jan-Christoph

Arch. Math., Brno 38, No. 4, 307-309 (2002).

Summary: Hadamard’s classical gap theorem states that if \(f (z) = \sum ^{\infty }_{l = 1} a_l z^{k_l}\) is a power series with radius of convergence \(1\), and \(\frac {k_l + 1}{k_l} \geq \theta > 1\), the circle \(| x| = 1\) is the natural boundary of \(f\). By combining Turán’s proof of Fabry’s gap theorem with a gap theorem of P. Szüsz we obtain a gap theorem which is more general then both these theorems.

MSC:

30A10	Inequalities in the complex plane
30B10	Power series (including lacunary series) in one complex variable
11N30	Turán theory

Keywords:

gap theorems; lacunary series; power sum method