A heuristic principle for a nonessential isolated singularity

Abstract

A heuristic principle in function theory claims that a family of holomorphic (meromorphic) functions which share a property $P$ in a region $\Omega$ is likely to be normal in $\Omega$ if $P$ cannot be possessed by nonconstant entire (meromorphic) functions in the finite plane. L. Zalcman established a rigorous version of this principle. An analogous principle for a nonessential singularity is plausible: If a holomorphic (meromorphic) function $f$ has an isolated singularity at ${z_0}$, and in a deleted neighborhood of ${z_0}$ the function $f$ has a property $P$ which cannot be possessed by nonconstant entire (meromorphic) functions in the finite plane, then ${z_0}$ is a nonessential singularity. We establish a rigorous version of the principle for holomorphic functions that is very similar to Zalcman’s precise statement of the other principle. However, this rendition of the heuristic principle for a nonessential singularity fails for meromorphic functions in contrast to Zalcman’s solution.