Inequalities for Special Strong Differential Superordinations Using a Generalized Sălăgean Operator and Ruscheweyh Derivative

Abstract

In the present paper we establish several inequalities for strong differential superordinations regarding the extended new operator $RD_{\lambda ,\alpha }^{m}$ defined by using the extended Sălăgean operator and the extended Ruscheweyh derivative, $RD_{\lambda ,\alpha }^{m}: \mathscr {A}_{n\zeta }^{\ast }\rightarrow \mathscr {A}_{n\zeta }^{\ast },$ $RD_{\lambda ,\alpha }^{m}f(z,\zeta )=(1-\alpha )R^{m}f(z,\zeta )+\alpha D_{\lambda }^{m}f(z,\zeta ),$ z ∈ U, $\zeta \in \overline {U},$ where R ^mf(z, ζ) denote the extended Ruscheweyh derivative, $D_{\lambda }^{m}f(z,\zeta )$ is the extended generalized Sălăgean operator, and $\mathscr {A}_{n\zeta }^{\ast }=\{f\in \mathscr {H}(U\times \overline {U}),\ f(z,\zeta )=z+a_{n+1}\left ( \zeta \right ) z^{n+1}+\dots ,\ z\in U,$ $\zeta \in \overline {U}\}$ is the class of normalized analytic functions.