The Julia-Caratheodory theorem on the bidisk revisited

TL;DR

This paper revisits the Julia-Carathéodory theorem on the bidisk, establishing conditions under which functions have linear directional derivatives based on boundary behavior of the Julia quotient.

Contribution

It proves that uniform boundedness of the Julia quotient along all nontangential approaches ensures linearity of directional derivatives, extending prior results.

Findings

Uniform boundedness implies linear directional derivatives.

Lipschitz condition near boundary yields linear derivatives for rational functions.

Refined understanding of boundary behavior in the bidisk context.

Abstract

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carath\'eodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.