On Wiman's theorem concerning the minimum modulus of a~function analytic in the unit disk

O. B. Skaskiv

Izvestiya. Mathematics , Tome 35 (1990) no. 1, pp. 165-182

Résumé

This paper contains an investigation of conditions under which an analytic function $F(z)$ represented by a Dirichlet series $$ F(z)=\sum_{n=0}^\infty a_ne^{z\lambda_n},\qquad 0=\lambda_0\lambda_n\uparrow+\infty\quad(n\to+\infty), $$ absolutely convergent in $\{z\colon\operatorname{Re}z0\}$ satisfies the relation $$ F(x+iy)=(1+o(1))a_{\nu(x)}e^{(x+iy)\lambda_{\nu(x)}} $$ uniformly with respect to $y\in\mathbf R$ as $x\to-0$ in the complement of some sufficiently small set. The results are used to derive as simple corollaries new assertions for functions analytic in the unit disk that are represented by lacunary power series. All the assertions proved in this article are best possible or close to best possible. Bibliography: 12 titles.