A question of Gol'dberg concerning entire functions with prescribed zeros
The result of the paper under review is as follows. Let \(\varphi(r)>0\) be a non-decreasing function as \(r\to\alpha\), such that \(\int^ \infty(\varphi(t)t\log t)^{-1} dt< \alpha\), and \(\{z_ j\}\), \(1<\alpha\), \(| z_ j|\to \alpha\), be a sequence of complex numbers with a counting function \(n(r)\); suppose \[ 0< \liminf_{r\to \alpha}\log n(r)/\log r\leq \alpha. \] Then there exists an entire function \(f\) with the zeros \(z_ j\) such that \[ \log M(r,f)= o((\log n(r))^ 2 \varphi(\log n(r))),\quad r\to \alpha, \] outside of some exceptional set of finite logarithmic measure. The integral condition on \(\varphi\) can not be improved.
scientific article; zbMATH DE number 569418