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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On an extremal property of Doob's class


Author: J. S. Hwang
Journal: Trans. Amer. Math. Soc. 252 (1979), 393-398
MSC: Primary 30D99
MathSciNet review: 534128
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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\vert {f({P_n})} \right\vert$, 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\Vert f \right\Vert\, = \,{2 /e}$, where $ \left\Vert f \right\Vert\, = \,{\sup _{z \in U}}(1\, - \vert z{\vert^2})\vert f'(z)\vert$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0534128-6
Keywords: Extremal property and Doob's class
Article copyright: © Copyright 1979 American Mathematical Society