Extension Operators that Preserve Geometric and Analytic Properties of Biholomorphic Mappings

In this survey we are concerned with certain extension operators which take a univalent function f on the unit disc U to a univalent mapping F from the Euclidean unit ball B n in ℂ n $$\mathbb{C}^{n}$$ into ℂ n $$\mathbb{C}^{n}$$ , with the property that f ( z 1 ) = F ( z 1 , 0 ) $$f(z_{1}) = F(z_{1},0)$$ . This subject began with the Roper–Suffridge extension operator, introduced in 1995, which has the property that if f is a convex function of U then F is a convex mapping of B n . We consider certain generalizations of the Roper–Suffridge extension operator. We show that these operators preserve the notion of g-Loewner chains, where g ( ζ ) = ( 1 − ζ ) ∕ ( 1 + ( 1 − 2 γ ) ζ ) $$g(\zeta ) = (1-\zeta )/(1 + (1 - 2\gamma )\zeta )$$ , | ζ |