arXiv subjects

Andreas Sauer

Publications and source records attributed to Andreas Sauer.

4 recordsLinked to original sources

arXiv2025

Rational functions that share finite values with their first derivative

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

math.CV↗

arXiv2025

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.

math.CV↗

arXiv2025

Meromorphic functions that partially share values with their first derivative

We consider uniqueness results for meromorphic functions $f:{\mathbb C} \to \widehat{\mathbb C}$ such that for certain values $a\in {\mathbb C}$ the implication $f(z)=a \Rightarrow f'(z)=a$ holds, i.e. that $f$ and $f'$ share values {\it partially}. In particular, we give a result for four partially shared values.

math.CV↗

arXiv2022

A uniqueness problem concerning entire functions and their derivatives

We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a normality criterion, which might be interesting in its own right.

math.CV↗