Minimum modulus of lacunary power series and h-measure of exceptional sets

Abstract

We consider some generalizations of Fenton theorem for the entire functions represented by lacunary power series. Let f(z) = ∑︀+∞ k=0 fkz nk , where (nk) is a strictly increasing sequence of non-negative integers. We denote by Mf (r) = max{|f(z)| : |z| = r}, mf (r) = min{|f(z)| : |z| = r}, μf (r) = max{|fk|rk : k > 0} the maximum modulus, the minimum modulus and the maximum term of f, respectively. Let h(r) be a positive continuous function increasing to infinity on [1,+∞) with a nondecreasing derivative. For a measurable set E ⊂ [1,+∞) we introduce h − meas (E) = ∫︀ E dh(r) r . In this paper we establish conditions guaranteeing that the relations Mf (r) = (1 + o(1))mf (r), Mf (r) = (1 + o(1))μf (r) are true as r → +∞ outside some exceptional set E such that h − meas (E) < +∞. For some subclasses we obtain necessary and sufficient conditions. We also provide similar results for entire Dirichlet series.