Covering theorems for meromorphic functions
Let f be a meromorphic function in a domain \({\mathcal D}\subset {\bar {\mathbb{C}}}\). Denote by n(w) the number card \(f^{-1}(w)\), \(w\in {\bar {\mathbb{C}}}\) (counting multiplicities). The function f is p-valent if \(\max_{w}n(w)=p\). The author studies some topological properties of p- valent functions. For example: If f is p-valent in the unit disk then the set \(\{\) w: n(w)\(\leq p-1\}\) contains a non-degenerate continuum. If in addition the total branch order \(\neq 2p-2\) then \(n(w_ 0)\leq p-2\) for at least one point \(w_ 0\in {\bar {\mathbb{C}}}\).
scientific article; zbMATH DE number 3906842