2018/09/24 by Pascoe, J. E., Sargent, Meredith, Tully-Doyle, Ryan
#47A10 Secondary 47A55 #47A57 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 30E20
paper · doi:10.48550/arxiv.1809.09208
Let f: D → Ω be a complex analytic function. The Julia quotient is given by the ratio between the distance of f(z) to the boundary of Ω and the distance of z to the boundary of D. A classical Julia-Carathéodory type theorem states that if there is a sequence tending to τ in the boundary of D along which the Julia quotient is bounded, then the function f can be extended to τ such that f is nontangentially continuous and differentiable at τ and f(τ) is in the boundary of Ω. We develop an extended theory when D and Ω are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called λ-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using γ-regularity. Applications are given, including perturbation theory and moment problems.