Entire functions sharing simple $a$-points with their first derivative II

Abstract

We discuss some results around the following question: Let $f$ be a nonconstant complex entire function and $a$, $b$ two distinct complex numbers. If $f$ and its derivative $f^{\prime }$ share their simple $a$-points and also share the value $b$, does this imply $f\equiv f^{\prime }$?

11 pages; comments welcome