Variability Regions for Bounded Analytic Functions with Applications to Families Defined by Subordination Article Swipe

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Unit disk Subordination (linguistics) Mathematics Bounded function Analytic function Riemann zeta function Class (philosophy) Boundary (topology) Combinatorics Pure mathematics Mathematical analysis Computer science Philosophy Linguistics Artificial intelligence

Yusuf Abu-Muhanna , Thomas H. MacGregor ·

YOU? · · 1980 · Open Access · · DOI: https://doi.org/10.2307/2042952 · OA: W4242353200

We examine the set of points $(\varphi (\zeta ),\varphi â(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))$ 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 \[ \max \operatorname {Re} F(\varphi (\zeta ),\varphi â(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))\] where $|\zeta | < 1$ and F is analytic.