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Certain applications of differential subordination. (English) Zbl 0608.30013

Let A denote the class of functions f that are regular in the unit disk E with \(f(0)=0=f'(0)-1\). For real a let \(k_ a(z)=z/(1-z)^ a\) and let h be a regular convex univalent function with \(h(0)=1\) and \(Re[h(z)]>0\) in E. The authors define classes \(K_ a(h)\) to be the set of \(f\in A\) such that \(1+z(k_ a*f)'(z)/(k_ a*f)(z)\) is subordinate to h in E. Various properties of \(K_ a(h)\) and certain related classes are studied. These classes generalize various subclasses of A such as the class of \(\alpha\)- convex functions. See, for example, S. S. Miller, P. T. Mocanu and M. O. Reade [Rev. Roum. Math. Pures Appl. 19, 213-224 (1974; Zbl 0278.30011)].
Reviewer: D.V.V.Wend

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

Citations:

Zbl 0278.30011
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