Conformal mapping of parabola-shaped domains (Q1880506)

Let \(P=\{z=x+iy:x>0,\;| y| <Ax^\alpha\}\), \(0<\alpha<1\), be a parabola-shaped domain and \(S=\{w+u+iv:| v| <\pi/2\}\) be a strip in the complex plane. Suppose that \(f\) is the conformal mapping which maps \(P\) onto \(S\) and is real and increasing for real \(z\). The authors obtain sharp estimates for \(f\) and \(f^\prime\). Precisely, they prove the asymptotic formulae \[ f(z)=\frac{\pi}{2A(1-\alpha)} z^{1-\alpha}+o(1), \] and \[ f^\prime(z)=\left [\frac{\pi}{2A}+o(1)\right ]x^{-\alpha} \] uniformly as \(z\to \infty\) in \(P\). Classical results for more general conformal maps due to Lindelöf and Warschawski are used in the proof. However, in addition, the authors use long arguments associated with the specific map \(f\) to obtain the above asymptotic formulae. These formulae are used in the paper ``Sharp integrability for Brownian motion in parabola-shaped domains'' [to appear in J. Funct. Anal.] by \textit{R. Bañuelos} and \textit{T. Carroll} for the study of the exponential decay of harmonic measure in parabola-shaped domains in the \(n\)-dimensional Euclidean space.