Rational functions that share finite values with their first derivative

Abstract

We treat shared value problems for rational functions $R(z)$ and their derivative $R^{\prime }(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and ${\partial }_{z}R=\hat{I}»w\cdot R^{\prime }(w)$ on $\mathbb{C}\setminus \{0\}$ and $\hat{\mathbb{C}}$, again with rational functions $R$. In $\mathbb{C}\setminus \{0\}$ this is related to shared values of meromorphic functions $f:\mathbb{C}\to \hat{\mathbb{C}}$ and $f^{\prime }$ through $f(z)=R(w)$ with $w=\exp (\hat{I}»z)$, while on $\hat{\mathbb{C}}$ this is connected to shared limit values in a similar fashion.

28 pages