Recently I come cross a question about deficient values of entire functions.

I find that many examples in the book about functions $f$ whose deficient values are singularities of the inverse $f^{-1}$.

I want to know whether there exist an example with the following property (in some sense it aks whether the concept of deficient value has a geometric explanation):

Is there an entire function $f$ with deficient value $a$ such that $a$ is not in the closure of the singular value set?

Another question: For entire functions in the Eremenko-Lybuich class, is it true that the the set of finite difficiency value is also bounded?

Any comments will be appreciated.