A remark concerning normal families and shared values

Abstract

We improve well-known results concerning normal families and shared values of meromorphic functions in the plane. In particular, we obtain two corollaries concerning meromorphic functions $f:\mathbb{C}\to \hat{\mathbb{C}}$: i) If $f$ shares a non-zero finite value with $f^{\prime }$, and such that $f^{\prime }$ is bounded on the preimages of $f$ for a second value, then $f$ is normal. ii) If $f$ shares two finite values with $f^{\prime }$, then $f$ and $f^{\prime }$ are normal.

8 pages