Extension operators and automorphisms

Abstract

Many of the commonly studied extension operators that take a normalized locally univalent function f defined in the open unit disk D ⊆ C to a normalized locally biholomorphic mapping of the unit ball B ⊆ C n , n ≥ 2 , are of the form f ↦ Ψ ∘ Φ 0 [ f ] , where Φ 0 [ f ] ( z ) = ( f ( z 1 ) , f ′ ( z 1 ) z ˆ ) , z ∈ B , z ˆ = ( z 2 , … , z n ) , is the Roper–Suffridge extension of f and Ψ is a first-coordinate shear, an automorphism of C n of the form Ψ ( w ) = ( w 1 + g ( w ˆ ) , w ˆ ) , where g : C n − 1 → C is holomorphic and contains no terms of degree 0 or 1 in its Taylor expansion at the origin. Other studies have considered the Roper–Suffridge extension operator with the multiplier f ′ ( z 1 ) replaced by a different expression. Recently, it has been shown that the set of restrictions of automorphisms of C n to B satisfying some geometric property such as convexity or starlikeness is dense in the full respective family of geometric mappings when n ≥ 2 . We study large families of biholomorphic mappings on B generated by automorphisms of C n , abstract generalizations of the Roper–Suffridge extension operator, unitary rotations, and families of locally univalent functions on D . In many cases, families arising from distinct automorphisms will be seen to be disjoint, despite the fact that they contain the restrictions of the respective automorphisms to B and the previously noted density result. Of particular interest given our consideration of more general automorphisms related to extension operators, we show that the compositions of the Roper–Suffridge extension operator with a particular class of first-coordinate overshears, automorphisms of C n of the form Ψ ( w ) = ( h ( w ˆ ) w 1 + g ( w ˆ ) , w ˆ ) for g , h : C n − 1 → C holomorphic and h never 0, extend certain bounded starlike mappings of D to starlike mappings of B . We conclude with several open questions inspired by this work.