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

摘要

Let a fractional operator D_λ~(n, α) (n ∈ N_0 = {0, 1, 2, ...}, 0 ≤ α < 1, λ ≥ 0) be defined byD_λ~(0, 0) = f (z),D_λ~(1, α) f (z) = (1 - λ) Ω~α f (z) + λ z (Ω~α f (z))~′ = D_λ~α (f (z)),D_λ~(2, α) f (z) = D_λ~α (D_λ~(1, α) f (z)),?D_λn, α f (z) = D_λ~α (D_λ~(n - 1, α) f (z)), whereΩα f (z) = Γ (2 - α) z~α D_z~α f (z), and D_z~α 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 D_λ~(n, α), which yield sharp distortion, rotation theorems and Koebe domain. These results extended corresponding previously known results.