On a conjecture of W. Bergweiler

Abstract

In this paper it is discussed for which meromorphic functions $f$ the homogeneous differential $f(z) f''(z) - a(f'(z))^2$ has only finitely many zeros. It is shown that any transcendental meromorphic functions $f(z)$ have the form $R(z) \exp(P(z))$ for a rational function $R$ and a polynomial $P$ with the property if $a \ne 1, (n \pm 1) / n$, $n \in N$. This result settles one conjecture proposed by W. Bergweiler.