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

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