The growth irregularity of slowly growing entire functions

Abstract

We show that entire transcendental functions f satisfying log M(r,f) = o(log 2r), r → ∞ (M(r,f): = maxf(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 pr) +0 (log2-pr), r → ∞ is impossible. It becomes possible if "o" is replaced by "O.". © Springer Science+Business Media, Inc. 2006.