Modified Roper-Suffridge operator for some holomorphic mappings

Abstract

In this paper, we will study the operator given by $$F(z) = (f(z_1 ) + f'(z_1 )P(z_0 ),(f'(z_1 ))^{1/k} z_0 ^T )^T ,$$ where z = (z1, z0T)T belongs to the unit ball Bn in ℂn, z1 ∈ U = B1, z0 = (z2, …, zn)T ∈ ℂn−1, and P: ℂn−1 → ℂ is a homogeneous polynomial of degree k (k ⩾ 2), the holomorphic branch is chosen such that (f′(0))1/k = 1. We will give different conditions for P such that the modified operator preserves the properties of almost spirallikeness of type β and order α, spirallikeness of type β and order α, and strongly spirallikeness of type β and order α, respectively.