Subordination and superordination for certain analytic functions associated with Ruscheweyh derivative and a new generalised multiplier transformation

Abstract

In the present paper, we study the operator defined by using Ruscheweyh derivative $\mathcal{R}^m$ and new generalized multiplier transformation $$ \mathcal{D}^{m}_{\lambda_{1},\lambda_{2},\ell,d }f(z) =z+\sum_{k=n+1}^{\infty}\left[\dfrac{\ell(1+(\lambda_{1}+\lambda_{2})(k-1))+d}{\ell(1+\lambda_{2}(k-1))+d}\right]^m a_kz^{k}$$ denoted by $\mathcal{R}\mathcal{D}^{m,\alpha}_{\lambda_{1},\lambda_{2},\ell,d }:\mathcal{A}_n\rightarrow \mathcal{A}_n$, $ \mathcal{R}\mathcal{D}^{m,\alpha}_{\lambda_{1},\lambda_{2},\ell,d }f(z)=(1-\alpha) \mathcal{R}^mf(z)+ \alpha\mathcal{D}^{m}_{\lambda_{1},\lambda_{2},\ell,d }f(z) $, where $ \mathcal{A}_{n}=\left\{f\in \mathcal{H}(\mathbb{U}), f(z) =z+a_{n+1}z^{n+1} +a_{n+2}z^{n+2}+...,z\in\mathbb{U}\right\}$ is the class of normalized analytic functions with $\mathcal{A}_{1}=\mathcal{A}$. We obtain several differential subordinations associated with the operator $\mathcal{R}\mathcal{D}^{m,\alpha}_{\lambda_{1},\lambda_{2},\ell,d }f(z)$. Further, sandwich-type results for this operator are considered.