It seems that there may be example of a transcendental entire function with finite (but positive) planar area of the Fatou set in Eremenko-Lyubich class. However, I can't not find it in the literature for Eremenko-Lyubich class. I remember that I may see a related title without any further details in the lecture notes of Christopher Bishop of transcendental 
dynamical system, however, I can not find it in the update version of the notes
from his homepage.


I do not know whether such example have already constructed yet beyond Eremenko-Lyubich class. (I Guess it may been constructed by the method used in the paper of Eremenko and Lyubich of Examples of entire functions with pathological dynamics.)


The comments and remarks will be appreciated.



Is there an example with Area $0<F(f)<\infty$ for some transcendental entire function
Yes there are such examples, the simplest one is $\sin(z^3)$.
It has a stronger property that the area of non-escaping set is finite.
MR2213937 
Hemke, Jan-Martin,
Recurrence of entire transcendental functions with simple post-singular
sets. 
Fund. Math. 187 (2005), no. 3, 255–289.


