Analytic and geometric properties associated with some extension operators

Abstract

In this article we are concerned with some extension operators Φ n,α and Φ n,Q that provide a way of extending a locally univalent function f on the unit disc U to a locally biholomorphic mapping F ∈ H(B n ). By using the method of Loewner chains, we prove that if f can be embedded as the first element of a g-Loewner chain on the unit disc, where for |ζ| < 1 and γ ∈ (0, 1), then F = Φ n,α(f) can also be embedded as the first element of a g-Loewner chain on B n , whenever . In particular, if f is star-like of order γ on U (resp. f is spiral-like of type β and order γ on U, where β ∈ (−π/2, π/2)), then F = Φ n,α(f) is also star-like of order γ on B n (resp. F = Φ n,α(f) is spiral-like of type β and order γ on B n ). Also, if f is almost star-like of order β and type γ on U, where β ∈ [0, 1), then F = Φ n,α(f) is almost star-like of order β and type γ on B n . Similar ideas are applied in the case of the Muir extension operator Φ n,Q , where Q is a homogeneous polynomial of degree 2 on ℂ n−1 such that and γ ∈ (0, 1).