Polynomially bounded solutions to the Loewner differential equation in several complex variables

Abstract

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f (z, t) = e t A z + · · · , where A ∈ L(C n , C n ) has the property m( A) > 0. Here m( A) = min{ A(z), z : z = 1}. We also give sufficient conditions for g(z, t) = L( f (z, t)) to be polynomially bounded, where f (z, t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.

Univalent Subordination Chains in Reflexive Complex Banach Spaces

“…is the Carathéodory class. For various applications of these classes in the theory of biholomorphic mappings in C n and complex Banach spaces, see [15], [20], [23], [24], [25], [27], [31], [32], [33], [42], [47], [50].…”

“…In this paper, we consider the notions of A-normalized univalent subordination chain and A-parametric representation on the unit ball of a reflexive complex Banach space X, where A ∈ L(X) satisfies the condition k + (A) < 2m(A). Note that A-normalized univalent subordination chains have been recently investigated in [24], in the case X = C n (see also [2], [31] and [51]).…”

“…We next consider the connection between any standard solution of (3.26) and the A-normalized univalent subordination chain given by (3.14) (see [15,Theorem 3.1], [29,Theorem 2.4], [31,Theorem 3.3], in the case X = C n ; compare [2], [4], [31], [51]). First, we prove the following result.…”

Convergence results for families of univalent mappings on the unit ball in C^n

“…Since the early works devoted to Loewner chains and the Loewner differential equation in higher dimensions due to Pfaltzgraff [27] and Poreda [28,29], many results in this field have been obtained (see [1,5,6,9,11,13,14,15,20,21,35]). We also mention the main contributions of Bracci [5] related to the existence of bounded support points for the family S 0 (B n ), n ≥ 2, and of Roth [31] concerning the ndimensional version of the well known Pontryagin maximum principle.…”

On Reachable Families of the Loewner Differential Equation in Several Complex Variables

“…For a study of the Loewner equations in several complex variables, regarding the general condition m(A) > 0, one can consult [1,17,38]. Definition 1.1 (see [5,11,13] This family represents a generalization of the well known Carathéodory family P, of normalized holomorphic functions on the unit disc with positive real part.…”