Area of the complement of the fast escaping sets of a family of entire functions

Abstract

Let f be an entire function with the form f(z) = P(e^z)/e^z, where P is a polynomial with deg(P) ≥ 2 and P(0) ≠ 0. We prove that the area of the complement of the fast escaping set (hence the Fatou set) of f in a horizontal strip of width 2π is finite. In particular, the corresponding result can be applied to the sine family α sin(z + β), where α ≠ 0 and β ∈ C.