Optimization and estimates of harmonic measure (Q1106981)

Let D be a connected set towards \(\infty\) in the plane and \(\theta (r)=| \theta:\) \(re^{i\theta}\in D|\). For functions defined in D or taking values in D there are well-known results especially when D is ``narrow'' in some sense. This fact is here expressed by means of an integral inequality involving a relatively small \(\psi\) (R), for instance \(\int^{R}_{1}t\theta (t) dt\leq 2\pi \psi (R),\) \(R\geq 1\). A certain harmonic measure is essential for the applications in function theory. An estimate of this harmonic measure involves \(\inf \int^{R}_{1}\theta (t)^{-1} dt/t,\) where intervals in (1,R) with \(\theta (t)=2\pi\) should be taken away. The main part of the present paper is devoted to the rather complicated finding of this lower bound for \(\theta\) satisfying the integral inequality. As exmaples are given theorems about functions in the unit circle, about harmonic majorants of subharmonic functions, espcially a theorem of Phragmén-Lindelöf type, and about ``quadrature domains''. A misprint: in formula (7.5), write \(t\theta\) (t) instead of \(\theta\) (t)!.