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 degree at least $2$ and $P(0)\ne 0$. 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 $2\ddot{I}€$ is finite. In particular, the corresponding result can be applied to the sine family $\hat{I}\pm \sin (z+{\hat{I}}^2)$, where $\hat{I}\pm \ne 0$ and ${\hat{I}}^2\in \mathbb{C}$.

18 pages, 1 figure; to appear in Kodai Math. J