Extension operators for locally univalent mappings

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: