Functions holomorphic along a $C^1$ pencil of holomorphic discs

TL;DR

This paper generalizes Forelli's theorem to functions holomorphic along a broad class of holomorphic discs, extending previous results and resolving an open problem in the field.

Contribution

It introduces a generalized version of Forelli's theorem applicable to functions holomorphic along a $C^1$ pencil of holomorphic discs, broadening the theorem's scope.

Findings

Generalization of Forelli's theorem for a $C^1$ pencil of holomorphic discs

Answers an open problem from Chirka (2006)

Extends previous results by JKS13

Abstract

The main purpose of this article is to present a generalization of Forelli's theorem for the functions holomorphic along a general pencil of holomorphic discs. This generalizes the main result of \cite{JKS13} and the original Forelli's theorem, and furthermore, answers one of the problems posed in \cite{Chirka06}.