Unbounded domains of normality of entire functions of small growth

Abstract

Let f: C → C denote a transcendental entire function and let m(r, f) equal min {[mid ]f(z)[mid ][ratio ][mid ]z[mid ] = r}. In this note, we mainly prove that ifformula herethen the set of normality N(f) of f has no unbounded preperiodic and periodic components. In particular, f has no Baker domains. We also discuss non-existence of unbounded components of normality under other conditions.