## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Normal families of holomorphic functions and multiple values

#### Abstract

Let $\mathcal{F}$ be a family of holomorphic functions defined in $D \subset C$, and let $k, m, n, p$ be four positive integers with $\frac{k+p+1}{m}+\frac{p+1}{n} < 1$. Let $\psi (\not \equiv 0, \infty )$ be a meromorphic function in $D$ and which has zeros only of multiplicities at most $p$. Suppose that, for every function $f \in \mathcal{F}$, (i) $f$ has zeros only of multiplicities at least $m$; (ii) all zeros of $f^{(k)}-\psi(z)$ have multiplicities at least $n$; (iii) all poles of $\psi$ have multiplicities at most $k$, and (iv) $\psi(z)$ and $f(z)$ have no common zeros, then $\mathcal{F}$ is normal in $D$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3 (2012), 535-547.

Dates
First available in Project Euclid: 14 September 2012