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Zbl 1021.30022
Nunokawa, Mamoru; Owa, Shigeyoshi; Takahashi, Norihiro; Saitoh, Hitoshi
Sufficient conditions for Carathéodory functions.
(English)
[J] Indian J. Pure Appl. Math. 33, No.9, 1385-1390 (2002). ISSN 0019-5588; ISSN 0975-7465/e

Let $A$ denote the class of functions $f(z)= 1+p_1z+ p_2z^2\dots$ analytic in the unit disk $U$, $p\in A$ with $\operatorname {Re} p(z)> 0$ for $z\in U$ is a Carathéodory function. In this note, the authors prove some sufficient conditions for $f\in A$ to be a Carathéodory function, in terms of differential subordinations. The proofs are based on a result of {\it S. S. Miller} and {\it P. T. Mocanu} [J. Math. Anal. Appl. 65, 289-305 (1978; Zbl 0367.34005)].
[O.Fekete (Freiburg)]
MSC 2000:
*30C80 Maximum principle, etc. (one complex variable)
34A40 Differential inequalities (ODE)

Keywords: differential subordination; Carathéodory functions

Citations: Zbl 0367.34005

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