Fournier, Richard The range of a continuous linear functional over a class of functions defined by subordination. (English) Zbl 0715.30010 Glasg. Math. J. 32, No. 3, 381-387 (1990). 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 Cited in 4 Documents 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 Keywords:subordinate; Hadamard product PDFBibTeX XMLCite \textit{R. Fournier}, Glasg. Math. J. 32, No. 3, 381--387 (1990; Zbl 0715.30010) Full Text: DOI References: [1] DOI: 10.1007/BF02566116 · Zbl 0261.30015 · doi:10.1007/BF02566116 [2] DOI: 10.2307/2040127 · Zbl 0311.30010 · doi:10.2307/2040127 [3] Hallenbeck, Linear problems and convexity techniques in geometric function theory (1984) · Zbl 0581.30001 [4] Ahlfors, Conformal invariants (1973) [5] Duren, Univalent functions (1983) [6] DOI: 10.2307/1995600 · Zbl 0227.30013 · doi:10.2307/1995600 [7] Fournier, Complex Variables 11 pp 125– (1989) · Zbl 0639.30016 · doi:10.1080/17476938908814330 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.