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

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
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