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 \(\Delta\).


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