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Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation. (English) Zbl 1132.30311
Summary: Let $\cal A$ denote the class of analytic functions with the normalization $f(0)=f'(0)-1=0$ in the open unit disk $\cal U$. Set $$ f^s_\lambda(z)=z+\sum^\infty_{k=2}\left(\frac{k+\lambda}{1+\lambda}\right)^s z^k\quad (s\in\Bbb R; \lambda>-1; z\in\cal U), $$ and define $f^s_{\lambda,\mu}$, in terms of the Hadamard product, $$ f^s_\lambda(z)* f^s_{\lambda,\mu}(z)=\frac{z}{(1-z)^\mu}\quad(\mu>0; z\in\cal U). $$ In this paper, the authors introduce several new subclasses of analytic functions defined by means of the operator $$I^s_{\lambda,\mu}:\cal A\to\cal A$$ given by $$ I^s_{\lambda,\mu}f(z)=f^s_{\lambda,\mu}(z)*f(z)\quad (f\in\cal A;\ s\in\Bbb R; \lambda>-1; \mu>0). $$ Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

MSC:
30C45Special classes of univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
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References:
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