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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
##### Keywords:
subordinate; $$\alpha$$-convex functions
Zbl 0278.30011
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