On uniqueness sets for areally mean 𝑝-valent functions

Abstract

We study the sets of uniqueness of areally mean p-valent functions in the unit disc. Namely, if f ( z ) f(z) is in this class and has the same angular limit in a set E on the boundary of the unit disc, we prove that if p is small compared to the size of E then f ( z ) f(z) is constant. We then construct an areally mean p-valent function which shows that some condition on the size of the set E must be imposed.