Abstract
For a transcendental entire function f, the property that there exists r>0 such that m^n(r)rightarrow infty as nrightarrow infty , where m(r)=min {|f(z)|:|z|=r}, is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).
Highlights
Let f be a transcendental entire function and denote by f n, n ∈ N, the n-th iterate of f
The results in this paper address a question in complex dynamics concerning the escaping set of a transcendental entire function f, defined as
There is a huge literature about the relationship between m(r ) and M(r ) for various types of transcendental entire functions
Summary
Let f be a transcendental entire function and denote by f n, n ∈ N, the n-th iterate of f. May or may not hold, depending on the function f It has been known for some time that the sequence Mn(r ) is of importance in relation to work on Eremenko’s conjecture, since it plays a key role in the definition of a subset of I ( f ) called the fast escaping set, all of whose components are unbounded; see, for example [12]. In [8] we obtained the following result, which gives a family of transcendental entire functions for which Eremenko’s conjecture holds in a strong way. Theorem 1.1 Let f be a real transcendental entire function of finite order with only real zeros for which there exists r > 0 such that mn(r ) → ∞ as n → ∞. Remark In [8, Thm. 1.1] we showed that, for such functions f , the set I ( f ) has the structure of an (infinite) spider’s web
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