On asymptotic values of entire functions

I. I. Marchenko

Izvestiya. Mathematics , Tome 63 (1999) no. 3, pp. 549-560

Résumé

In the theory of functions, an asymptotic spot of a function $f$ is defined to be a pair $\{a,\Gamma\}$, where $a\in\overline{\mathbb C}$ and $\Gamma$ is a continuous curve such that $f(z)\to a$ ($z\to\infty$, $z\in\Gamma$). In this paper we introduce a new notion of a strong asymptotic spot. Using this notion, we extend some results of Ahlfors (concerning asymptotic spots of entire functions of finite order) to functions of infinite order. We obtain exact estimates for the number of different strong asymptotic spots $\{\infty,\Gamma_j\}$ of entire functions of finite or infinite lower order.