The growth irregularity of slowly growing entire functions

Abstract

We show that entire transcendental functions f satisfying $$\log M(r,f) = o(\log ^2 r),r \to \infty (M(r,f): = \mathop {\max }\limits_{|z| = r} |f(z)|)$$ necessarily have growth irregularity, which increases as the growth diminishes. In particular, if 1 < p < 2, then the asymptotics $$\log M(r,f) = \log ^p r + o(\log ^{2 - p} r),r \to \infty ,$$ , is impossible. It becomes possible if “o” is replaced by “O.”