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The range of a continuous linear functional over a class of functions defined by subordination. (English) Zbl 0715.30010

Let \(\Delta =\{z\in {\mathbb{C}}:| z| <1\}\) and H(\(\Delta\)) be the set of analytic functions on \(\Delta\). Let \(F\in H(\Delta)\) be univalent, convex and bounded. Then F can be extended to a homeomorphism onto the closure of \(\Delta\). Let \(s(F)=\{f\in H(\Delta):\) f is subordinate to F in \(\Delta\}\) and for \(\theta\in {\mathbb{R}}\), let \(s(F,\theta)=\{f\in s(F):\lim_{z\to 1} f(z)=F(e^{i\theta})\}\). Let \(G\in H(\Delta)\) be convex and univalent in \(\Delta\) and let \[ I(f)=\int^{1}_{0}f*G(x)dx, \] where f*G denotes the Hadamard product of f and G. Theorem: For each \(\theta\in {\mathbb{R}}\), \(\{I(f):f\in s(F,\theta)\}=\{\frac{1}{z}\int^{z}_{0}F*G(x)dx:z\in \Delta\) or \(z=e^{i\theta}\). Moreover \[ I(f)=e^{- i\theta}\int^{e^{i\theta}}_{0}F*G(x)dx\Leftrightarrow f(z)\equiv F(e^{i\theta}z). \]
Reviewer: J.Waniurski

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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[1] DOI: 10.1007/BF02566116 · Zbl 0261.30015 · doi:10.1007/BF02566116
[2] DOI: 10.2307/2040127 · Zbl 0311.30010 · doi:10.2307/2040127
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