Some strong differential subordinations using a differential operator

Category: Vol 31/2015 no. 2 Tags: Strong differential subordination, Univalent function, Differential operator, Best dominant, Extended Ruscheweyh derivative, Extended generalized Sălăgean operato, Convex function

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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.