A New Roper-Suffridge Extension Operator on a Reinhardt Domain

and Applied Analysis 3 In contrast to the modified Roper-Suffridge extension operator in the unit ball, it is natural to ask if we can modify the Roper-Suffridge extension operator on the Reinhardt domains. In this paper, we will introduce the following modified operator: F z ⎛ ⎝f z1 f ′ z1 n ∑ j 2 ajz pj j , ( f ′ z1 2z2, . . . , ( f ′ z1 nzn ⎞ ⎠ ′ 1.5 on the Reinhardt domainΩn,p2,...,pn . Wewill give some sufficient conditions for aj under which the above Roper-Suffridge operator preserves an almost starlike mappings of order α and starlike mappings of order α, respectively. In the following, we give some notation and definitions. Let C be the space of n complex variables z z1, . . . , zn ′ with the Euclidean inner product 〈z,w〉 ∑n i 1ziwi and the Euclidean norm ‖z‖ 〈z, z〉, where z,w ∈ C and the symbol “′” means transpose. The unit ball of C is the set Bn {z ∈ C : ‖z‖ < 1}, and the unit sphere is denoted by ∂Bn {z ∈ C : ‖z‖ 1}. In the case of one complex variable, B1 is the unit disk, usually denoted by D. Let Ω be a domain in C. Denote H Ω by the space of all holomorphic mappings from Ω into C. A mapping f ∈ H Bn is called normalized if f 0 0 and Jf 0 In, where Jf 0 is the complex Jacobian matrix of f at the origin and In is the identity operator on C. A mapping f ∈ H Bn is said to be locally biholomorphic if det Jf z / 0 for every z ∈ Bn. A normalized mapping f ∈ H Bn is said to be convex if λω1 1−λ ω2 ∈ f Bn for arbitrary ω1, ω2 ∈ f Bn and 0 λ 1. A normalized mapping f ∈ H Bn is said to be starlike with respect to the origin if λf Bn ⊂ f Bn , 0 λ 1. A normalized mapping f ∈ H Bn is said to be ε starlike if there exists a positive number ε, 0 ε 1, such that f Bn is starlike with respect to every point in εf Bn . A domain Ω is called a Reinhardt domain if e1z1, e2z2, . . . , enzn ′ ∈ Ω holds for any z z1, z2, . . . , zn ′ ∈ Ω and θ1, θ2, . . . , θn ∈ R. A domain Ω is called a circular domain if ez ∈ Ω holds for any z ∈ Ω and θ ∈ R. The Minkowski functional ρ z of the Reinhardt domain Ωn,p2,...,pn ⎧ ⎨ ⎩ z ∈ C : |z1| n ∑ j 2 ∣ ∣zj ∣ ∣pj < 1 ⎫ ⎬ ⎭ , pj 1, j 2, . . . , n 1.6