Functional Analysis and Its Applications

, Volume 40, Issue 4, pp 304–312 | Cite as

The growth irregularity of slowly growing entire functions

  • I. V. Ostrovskii
  • A. E. Üreyen
Open Access


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.”

Key words

Clunie-Kövari theorem Erdös-Kövari theorem Hayman convexity theorem maximum term Levin’s strong proximate order 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • I. V. Ostrovskii
    • 1
    • 2
  • A. E. Üreyen
    • 1
  1. 1.Department of MathematicsBilkent UniversityAnkaraTurkey
  2. 2.Verkin Institute for Low Temperature Physics and EngineeringKharkovUkraine

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