A remark concerning normal families and shared values

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 $f \colon {\mathbb C} \to {\widehat{\mathbb C}}$: i) If $f$ shares a non-zero finite value with $f'$, and such that $f'$ is bounded on the preimages of $f$ for a second value, then $f$ is normal. ii) If $f$ shares two finite values with $f'$, then $f$ and $f'$ are normal.