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Neighborhoods and Carathéodory functions. (English) Zbl 0867.30009

Summary: We give a criterion for an analytic function in the unit disc satisfying a neighborhood inequality of Ruscheweyh type to be a Carathéodory function. As applications we also derive the following: For normalized analytic functions \(f\) and \(g\) in the unit disc \(\Delta\) the conditions that \(\text{Re} f'(z)> \beta\) and \(\text{Re} g'(z)>\beta\), where \(\beta=1- 1/[2 \sqrt{1-\ln 2}] \approx 0.097 \cdots\), imply \(f*g\) is starlike in the whole of \(\Delta\).

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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