A classical Julia-Carathéodory theorem states that if there is a sequence tending to $$\tau$$ in the boundary of a domain $$D$$ along which the Julia quotient is bounded, then the function $$\phi$$ can be extended to $$\tau$$ such that $$\phi$$ is nontangentially continuous and differentiable at $$\tau$$ and $$\phi(\tau)$$ is in the boundary of $$\Omega$$.
We develop a theory in the case of Pick functions where we consider sequences that approach the boundary in a controlled tangential way, yielding necessary and sufficient conditions for higher order regularity. In this talk, I will discuss the proof, including some of the technical details involved: amortization of the Julia Quotient, $$\gamma$$-regularity, and $$\gamma$$-auguries. I will also speak about some applications, including moment theory and the fractional Laplacian.