$\lambda$-fractional properties of generalized Janowski functions in the unit disc

Abstract

For analytic function $f(z)=z+a_2z^2+\cdots$ in the open unit disc $\mathbb{D}$, a new fractional operator $\mbox{D}^\lambda f(z)$ is defined. Applying this fractional operator $\mbox{D}^\lambda f(z)$ and the principle of subordination, we give new proofs for some classical results concerning the class $\cal{S}_\lambda^*(A,B,\alpha)$ of functions $f(z)$.