Subordination results for classes of analytic functions related to conic domains defined by a fractional operator

Let a fractional operator Dλn,α (n∈N0={0,1,2,…}n∈N0={0,1,2,…}, 0⩽α<10⩽α<1, λ⩾0λ⩾0) be defined byDλ0,0=f(z),Dλ1,αf(z)=(1−λ)Ωαf(z)+λz(Ωαf(z))′=Dλα(f(z)),Dλ2,αf(z)=Dλα(Dλ1,αf(z)),⋮Dλn,αf(z)=Dλα(Dλn−1,αf(z)), whereΩαf(z)=Γ(2−α)zαDzαf(z), and Dzα is the known fractional derivative. In this paper, several interesting subordination results are derived for certain classes of analytic functions related to conic domains defined by the operator Dλn,α, which yield sharp distortion, rotation theorems and Koebe domain. These results extended corresponding previously known results.