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

Abstract

Let a fractional operator D λ n , α ( n ∈ N 0 = { 0 , 1 , 2 , … } , 0 ⩽ α < 1 , λ ⩾ 0 ) be defined by D 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 λ n , α f ( z ) = D λ α ( D λ n − 1 , α f ( z ) ) , where Ω α 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.