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

Abstract

We show that if a complex entire function $f$ and its derivative $f'$ share their simple zeroes and their simple $a$-points for some nonzero constant $a$, then $f\equiv f'$. We also discuss how far these conditions can be relaxed or generalized. Finally, we determine all entire functions $f$ such that for 3 distinct complex numbers $a_1,a_2,a_3$ every simple $a_j$-point of $f$ is an $a_j$-point of $f'$.