Extension operators for locally univalent mappings

By Ian D. Graham, H. Hamada, G. Kohr, T. J. Suffridge. 2002

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A class of Loewner chain preserving extension operators (2008, methodology): “The Roper–Suffridge extension operator is defined by [Φ0,1/2(f)](z) = ( f(z1), √ f ′(z1) ẑ ) , f ∈ LS1, z ∈ B. (2.2) (Since f ′(∆) is a simply connected set that does not contain 0, a branch of √ f ′(·) may be chosen such that √ f ′(0) = 1.)”
Asymptotically spirallike mappings in several complex variables (2008, methodology): “We omit the proof of those assertions, since it suffices to use arguments similar to those in the proof of [ 11, Theorem 2.1 ].”