Geometric Function Theory in One and Higher Dimensions

Approximation properties of univalent mappings on the unit ball in Cn

“…If, in addition, D f (z) − I n ≤ c, z ∈ B n , for some c ∈ [0, 1), then f is quasiregular on B n and extends to a quasiconformal homeomorphism of C n onto itself (see [11,Lemma 2.2]; see e.g. [15,Chapter 8]). Note that Q(B n ) is a compact subset of H(B n ).…”

The Embedding Conjecture and the Approximation Conjecture in Higher Dimension

“…The class S 0 is compact [2]. We also denote with S(C n ) := {Ψ : C n −→ C n univalent : Ψ(0) = 0, (dΨ) 0 = Id} the space of normalized entire univalent functions.…”

“…We recall also the following result Proposition 1.1. [2] If (f t ) t≥0 is a normalized Loewner chain, then there exist an unique normalized biholomorphism Ψ : C n −→ R(f t ) and normal Loewner chain (g t ) t≥0 such that for each t ≥ 0 f t = Ψ • g t . In particular in the case t = 0, we have that if f ∈ S 1 then there exist Ψ ∈ S(C n ) and g ∈ S 0 such that f = Ψ • g. In particular, we have the following decomposition…”

“…The class S 0 is compact whereas S and S 1 are not [2,Chapter 8]. Hence, there are some normalized univalent functions on the ball that do not embed into a normal Loewner chain (i.e.…”

A subclass of close-to-convex functions

“…To prove the inequalities (8), let z = r e i θ (0 < r < 1). If ε denotes the closed line-segment in the complex ζ-plane from ζ = 0 and ζ = z, we have…”