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

Let $f$ be an entire function with the form $f(z)=P(e^z)\mathrm{/}{e}^z$, where $P$ is a polynomial with degree at least $2$ and $P(0)\mathrm{\ne }0$. 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 $2\pi$ is finite. In particular, the corresponding result can be applied to the sine family $\alpha \sin (z+\beta )$, where $\alpha \mathrm{\ne }0$ and $\beta \in \mathbb{C}$.