Growth conditions for entire functions with only bounded Fatou components
In this paper among others the following result is proved. Theorem: Let \(f\) be a transcendental entire function of order less than \(1/2\). Suppose that there exists \(\alpha> 1\), \(R_0> 1\), and a positive decreasing function \(\gamma(r)\) such that for every \(R> R_0\), there exists \(r\in(R,R^\alpha]\) with \[ {\log m(r,f)\over \log M(R,f)}\geq \alpha(1- \gamma(R)) \] and \(\sum_n \gamma(e^{2n})<\infty\). Then all components of the Fatou set of \(f\) are bounded. Some other interesting results concerning Fatou sets for entire function are obtained.
scientific article; zbMATH DE number 5624017
scientific article