The Growth Irregularity of Slowly Growing Entire Functions

I. V. Ostrovskii ; A. E. Üreyen

Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 4, pp. 72-82

Résumé

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

Détail

Keywords: Clunie–Kövari theorem, Erdös–Kövari theorem, Hayman convexity theorem, Levin's strong proximate order.
Mots-clés : maximum term