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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Hull subordination and extremal problems for starlike and spirallike mappings


Author: Thomas H. MacGregor
Journal: Trans. Amer. Math. Soc. 183 (1973), 499-510
MSC: Primary 30A32
MathSciNet review: 0338339
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Abstract: Let $ \mathfrak{F}$ be a compact subset of the family $ \mathcal{A}$ of functions analytic in $ \Delta = \{ z:\;\vert z\vert < 1\} $, and let $ \mathcal{L}$ be a continuous linear operator of order zero on $ \mathcal{A}$. We show that if the extreme points of the closed convex hull of $ \mathcal{F}$ is the set $ \{ {f_0}(xz)\} (\vert x\vert = 1)$, then $ \mathcal{L}(f)$ is hull subordinate to $ \mathcal{L}({f_0})$ in $ \Delta $. This generalizes results of R. M. Robinson corresponding to families $ \mathcal{F}$ of functions that are subordinate to $ (1 + z)/(1 - z)$ or to $ 1/{(1 - z)^2}$. Families $ \mathcal{F}$ to which this theorem applies are discussed and we identify each such operator $ \mathcal{L}$ with a suitable sequence of complex numbers.

Suppose that $ \Phi $ is a nonconstant entire function and that $ 0 < \vert{z_0}\vert < 1$. We show that the maximum of $ \operatorname{Re} \{ \Phi [\log (f({z_0})/{z_0})]\} $ over the class of starlike functions of order a is attained only by the functions $ f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;\vert x\vert = 1$. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0338339-9
PII: S 0002-9947(1973)0338339-9
Keywords: Subordination, hull subordination, continuous linear operator, functions with positive real part, Herglotz representation formula, extreme points, closed convex hull, univalent functions, starlike mapping, starlike functions of order a, spirallike functions, extremal problems, Kreĭn-Milman theorem, probability measure, convex mapping, convex functions of order a, continuous linear functional
Article copyright: © Copyright 1973 American Mathematical Society