carpathian_2015_31_2_143_156_001

Some strong differential subordinations using a differential operator

Andrei, Loriana and Choban, Mitrofan

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carpathian_2015_31_2_143_156

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In the present paper we study the operator RD_{\lambda ,\alpha}^{n}f(z,\zeta ) defined by using the extended Ruscheweyh derivative R^{n}f(z,\zeta ) and the extended generalized S\u{a}l\u{a}gean operator D_{\lambda }^{n}f(z,\zeta ), as RD_{\lambda ,\alpha }^{n}:\mathcal{A}_{\zeta }^{\ast }\rightarrow \mathcal{A}_{\zeta }^{\ast }, RD_{\lambda ,\alpha }^{n}f(z,\zeta )=(1-\alpha )R^{n}f(z,\zeta )+\alpha D_{\lambda}^{n}f(z,\zeta ), where \mathcal{A}_{n\zeta }^{\ast }=\{f\in \mathcal{H}(U\times \overline{U}),\ f(z,\zeta )=z+a_{n+1}\left( \zeta \right)z^{n+1}+\dots ,\ z\in U, \zeta \in \overline{U}\}\ is the class of normalized analytic functions with \mathcal{A}_{1\zeta }^{\ast }=\mathcal{A}_{\zeta }^{\ast }. We obtain several strong differential subordinations regarding the extended operator RD_{\lambda,\alpha }^{n}. Some examples are presented.

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Andrei, Loriana, Choban, Mitrofan

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