Extension operators for locally univalent mappings

Abstract

Let C denote the space ofn complex variables z = (z1, . . . , zn)with the Euclidean inner product 〈z,w〉 =∑n j=1 zj wj and the Euclidean norm ‖z‖ = 〈z, z〉1/2. Let z ′ = (z2, . . . , zn) so that z = (z1, z ′). Let B r = {z ∈ C : ‖z‖ < r} and let B = B 1 . In the case of one variable, B r is denoted by Ur and U1 by U. If G ⊂ C is an open set, let H(G) denote the set of holomorphic mappings from G into C. If f ∈H(Bn r ), we say that f is normalized if f(0) = 0 and Df(0) = I. Let S(B r ) be the set of normalized univalent mappings in H(B n r ). The sets of normalized convex (resp., starlike) mappings of B r are denoted by K(B n r ) (resp., S ∗(Bn r )). When n = 1, the sets S(U), S ∗(U), and K(U) are denoted by S, S ∗, andK, respectively. For vectors and matrices, A∗ denotes the conjugate transpose of A. We recall that a mapping F : B × [0,∞)→ C is called a Loewner chain if F(·, t) is univalent on B, F(0, t) = 0, DF(0, t) = eI for t ≥ 0, and F(z, s) ≺ F(z, t), z∈Bn, 0 ≤ s ≤ t <∞, where the symbol ≺ means the usual subordination. We will consider the set S 0(Bn) consisting of those mappings F ∈ S(B) that can be imbedded in Loewner chains. It is well known that, in the case of several complex variables, S 0(Bn) is a proper subset of S(B) (see [K; GrHK]). If F : B r → C (0 < r ≤ 1), we say that F ∈ S 0(Bn r ) if Fr ∈ S 0(Bn), where Fr(z) = r F(rz) and z∈Bn. A mapping f ∈H(Bn) with f(0) = 0 is called starlike if f is univalent on B and if f(B) is a starlike domain with respect to zero. It is known that starlikeness can be characterized in terms of Loewner chains: f is starlike on B iff f(z, t) = ef(z) (z ∈ B, t ≥ 0) is a Loewner chain. For the analytical characterization of starlikeness, see [S1; S2]. A key role in our discussion is played by the n-dimensional version of the Caratheodory set: