Growth conditions for entire functions with only bounded Fatou components |
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| Authors: | Aimo Hinkkanen Joseph Miles |
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| Affiliation: | 1.Department of Mathematics,University of Illinois at Urbana-Champaign,Urbana,USA |
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| Abstract: | Let f be a transcendental entire function of order less than 1/2. Denote the maximum and minimum modulus of f by M(r, f) = max{|f(z)|: |z| = r} and m(r, f) = min{|f(z)|: |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if | $
log log M(r,f) = O(log r/(log log r)^K )
$
log log M(r,f) = O(log r/(log log r)^K )
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