Article

    Ufa Mathematical Journal
    Volume 9, Number 4, pp. 135-144

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


    Salo T.M., Skaskiv O.B.

    DOI:10.13108/2017-9-4-135

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    We consider some generalizations of Fenton theorem for the entire functions represented by lacunary power series. Let f(z)=∑+∞k=0fkznk, where (nk) is a strictly increasing sequence of non-negative integers. We denote by Mf(r)=max 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,+\infty) with a non-decreasing derivative. For a measurable set E\subset [1,+\infty) we introduce h-\mathrm{meas}\,(E)=\int_{E}\frac{dh(r)}{r}. In this paper we establish conditions guaranteeing that the relations M_f(r)=(1+o(1)) m_f(r),\quad M_f(r)=(1+o(1))\mu_f(r) are true as r\to+\infty outside some exceptional set E such that h-\mathrm{meas}\,(E)<+\infty. For some subclasses we obtain necessary and sufficient conditions. We also provide similar results for entire Dirichlet series.