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

Anessa Oshah ; Maslina Darus

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2022), pp. 22-34.

Résumé

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 $\mathcal{R}{\mathcal{D}}_{{\lambda }_1,{\lambda }_2,\ell ,d}^{m,\alpha }:{\mathcal{A}}_n\to {\mathcal{A}}_n$, $\mathcal{R}{\mathcal{D}}_{{\lambda }_1,{\lambda }_2,\ell ,d}^{m,\alpha }f(z)=(1-\alpha ){\mathcal{R}}^{m}f(z)+\alpha {\mathcal{D}}_{{\lambda }_1,{\lambda }_2,\ell ,d}^{m}f(z)$, where ${\mathcal{A}}_n=\{f\in \mathcal{H}(\mathbb{U}),f(z)=z+a_{n+1}{z}^{n+1}+a_{n+2}{z}^{n+2}+...,z\in \mathbb{U}\}$ is the class of normalized analytic functions with ${\mathcal{A}}_1=\mathcal{A}$. We obtain several differential subordinations associated with the operator $\mathcal{R}{\mathcal{D}}_{{\lambda }_1,{\lambda }_2,\ell ,d}^{m,\alpha }f(z)$. Further, sandwich-type results for this operator are considered.

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