Functions with no unbounded Fatou components

Abstract

For a transcendental entire function with sufficiently small growth, Baker raised the question whether it has no unbounded Fatou components. We have shown that if the function is of order strictly less than half, minimal type, then it has no unbounded Fatou components. This, in particular gives a partial answer to Baker's question. In addition, we have addressed Wang's question on Fejér gaps. Certain results about functions with Fabry gaps and of infinite order have also been generalized.