λ-fractional properties of generalized Janowski functions in the unit disc

Abstract

For analytic function $f(z)=z+a_2{z}^2+\cdots$ in the open unit disc $\mathbb{D}$, a new fractional operator ${\text{D}}^{\lambda }f(z)$ is defined. Applying this fractional operator ${\text{D}}^{\lambda }f(z)$ and the principle of subordination, we give new proofs for some classical results concerning the class ${\mathcal{S}}_{\lambda }^{\mathcal{\ast }}\mathcal{(}\mathcal{A}\mathcal{,}\mathcal{B}\mathcal{,}\alpha \mathcal{)}$ of functions $f(z)$.