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Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. (English) Zbl 1132.30311

Summary: Let \(\mathcal A\) denote the class of analytic functions with the normalization \(f(0)=f'(0)-1=0\) in the open unit disk \(\mathcal U\). Set \[ f^s_\lambda(z)=z+\sum^\infty_{k=2}\left(\frac{k+\lambda}{1+\lambda}\right)^s z^k\quad (s\in\mathbb R; \lambda>-1; z\in\mathcal U), \] and define \(f^s_{\lambda,\mu}\), in terms of the Hadamard product, \[ f^s_\lambda(z)* f^s_{\lambda,\mu}(z)=\frac{z}{(1-z)^\mu}\quad(\mu>0; z\in\mathcal U). \]
In this paper, the authors introduce several new subclasses of analytic functions defined by means of the operator \[ I^s_{\lambda,\mu}:\mathcal A\to\mathcal A \] given by \[ I^s_{\lambda,\mu}f(z)=f^s_{\lambda,\mu}(z)*f(z)\quad (f\in\mathcal A;\;s\in\mathbb R; \lambda>-1; \mu>0). \]
Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

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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