Variability regions for bounded analytic functions with applications to families defined by subordination

Abu-Muhanna, Yusuf; MacGregor, Thomas H.

doi:10.1090/S0002-9939-1980-0577749-0

Variability regions for bounded analytic functions with applications to families defined by subordination
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by Yusuf Abu-Muhanna and Thomas H. MacGregor PDF

Proc. Amer. Math. Soc. 80 (1980), 227-233 Request permission

Abstract:

We examine the set of points $(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))$ where $|\zeta | < 1$ and $\varphi$ varies over the class of functions analytic in the open unit disk and is either (1) uniformly bounded or (2) subordinate to a given univalent function. In each case boundary points of the set correspond to unique functions associated with finite Blaschke products. This yields information about the form of solutions to extremal problems over the classes, including the problem \[ \max \operatorname {Re} F(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))\] where $|\zeta | < 1$ and F is analytic.

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