Parametric Representation of Univalent Mappings in Several Complex Variables

Abstract

Abstract. Let B be the unit ball of C n with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into C n of "positive real part", is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S 0 (B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr's work, we consider subsets of S 0 (B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S 0 (B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

Parametric representation and asymptotic starlikeness in ℂⁿ

“…The above condition is equivalent to the fact that there exists a Loewner chain f (z, t) such that {e −t f (·, t)} t≥0 is a normal family on B n and f (z) = f (z, 0), z ∈ B n (see [4], [7], [8]; cf. [16]).…”

“…These results show that the geometric properties of f (B n ) for f ∈ S(B n ) play an important role for the growth theorem. Another class of mappings for which the growth theorem holds is the class S 0 (B n ) (see [4], [7], [8], [9] for the Euclidean case and the case of a general norm, and [15], [16] for the case of the polydisc).…”

“…We remark that the above class plays a key role in the study of Loewner chains and the Loewner differential equation in dimension n ≥ 2 (see [4], [7], [8], [13]). [4], [7]; cf.…”

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

“…The following growth result was obtained by Graham, Hamada, and Kohr [11] for the family M (see [13,Lemma 1.2] in the case of mappings h ∈ N ; see also [34,Proposition 1.2.3] for the family N ). [8]; cf.…”

“…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.…”