  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.

Complex Variables