On an extremal property of Doob’s class

Abstract:

Recently, we have solved a long open problem of Doob (1935). To introduce the result proved here, we say that a function $f(z)$ belongs to Doob’s class D, if $f(z)$ is analytic in the unit disk U and has radial limit zero at an endpoint of some arc R on the unit circle such that $\operatorname {lim} {\operatorname {inf} _{n \to \infty }} \left | {f({P_n})} \right |$, where $\{ {P_n}\}$ is an arbitrary sequence of points in U tending to an arbitrary interior point of R. With this definition, our main result is the following extremal property of Doob’s class. Theorem. ${\operatorname {inf} _{f \in D}}\left \| f \right \| = {2 /e}$, where $\left \| f \right \| = {\sup _{z \in U}}(1 - |z{|^2})|f’(z)|$.