Functions of small growth with no unbounded Fatou components

Details Functions of small growth with no unbounded Fatou components Functions of small growth with no unbounded Fatou components

2008-01-23

arxiv.org/abs/0801.3610v1

Mathematics

Complex Variables

Scientific paper

We prove a form of the $\cos \pi \rho$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.

Affiliated with

Also associated with

No associations

Rating

Functions of small growth with no unbounded Fatou components does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Functions of small growth with no unbounded Fatou components, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Functions of small growth with no unbounded Fatou components will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-183874