On~an estimate for a~function represented by a~Dirichlet series

Z. Sh. Karimov

Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 525-532

Résumé

The article considers the series $$ f(z)=\sum_{k=1}^\infty a_ke^{\lambda_kz},\qquad0\lambda_k\uparrow\infty,\quad\sum_{k=1}^\infty\lambda_k^{-1}\infty, $$ convergent in the whole plane. Theorem 1. {\it Let $|f(x)|$ $-\infty$ where $0$. For given $\varepsilon>0$ and $h>0$ there exists a constant $A$, not depending on $f(z)$ and $H(x)$ such that $|f(z)|$ $x=\operatorname{Re}z,$ $|y|$.} Theorem 2. {\it If in addition $$ \delta=\varlimsup_{k\to\infty}\frac1{\lambda_k}\ln\biggl|\frac1{L'(\lambda_k)}\biggr|\infty,\qquad L(\lambda)=\prod_{k=1}^\infty\biggl(1-\frac\lambda{\lambda_k}\biggr), $$ then for arbitrary $z$ we have $|f(z)|$ $x=\operatorname{Re}z$.} The quantity $\delta$ cannot be replaced by a smaller one. These results strengthen corresponding results due to Gaier (RZhMat., 1967, 10B155) and Anderson and Binmore (RZhMat., 1972, 7B1115). Bibliography: 7 titles.