The size of an analytic function as measured by Lévy’s time change. (English) Zbl 0574.60087

McConnell, Terry R.

Ann. Probab. 13, 1003-1005 (1985).

The author proves that for a wide class of functions \(\Phi\), f analytic in the unit disk and \(| f'(0)| \geq 1\), it is true that \(E\Phi\) (\(\nu\) (f))\(\geq E(\Phi (\nu (z))\) where \(\nu (f)=\int^{\tau}_{0}| f'(B(s))|^ 2ds\), \(\tau\) being the exit time of the standard, plane, Brownian motion from the unit disk. Also the conjecture P(\(\lambda\leq \nu (f))\geq P(\lambda \leq \nu (z))\) is proved impossible in general.

Reviewer: H.Gzyl

MSC:

60J65

Brownian motion

Keywords:

complex Brownian motion; analytic functions; Brownian motion