Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$ C n

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

“…Next, we recall the notion of generalized parametric representation with respect to a time-dependent linear operator (see [22]; cf. Definition 2.4.…”

“…The authors in [22] have investigated various extremal properties of compact families S t A (B n ) (t ≥ 0) consisting of normalized biholomorphic mappings on the Euclidean unit ball B n in C n which have generalized parametric representation with respect to time-dependent linear operators A ∈ A. We have considered examples and applications which yield that the study of the family S t A (B n ) for time-dependent operators A ∈ A is basically different from that in the case of constant time-dependent linear operators (see [22]).…”

“…We have considered examples and applications which yield that the study of the family S t A (B n ) for time-dependent operators A ∈ A is basically different from that in the case of constant time-dependent linear operators (see [22]). In the case that A(t) = A, for all t ≥ 0, where A ∈ L(C n ) with k + (A) < 2m(A), then S t A (B n ) = S 0 A (B n ), for all t ≥ 0, where S 0 A (B n ) is the family of mappings with A-parametric representation (see [13]).…”

“…Note that k + (A) is the Lyapunov index of A and m(A) = min z =1 ℜ A(z), z . If n = 1 and a ∈ A, then S t a (U) = S, for all t ≥ 0 (see [22]), where S is the family of normalized univalent functions on the unit disc U.…”

“…The authors in [22] obtained extremal properties for the family S t A (B n ) consisting of normalized univalent mappings on B n which have generalized parametric representation with respect to time-dependent operators A ∈ A, and deduced certain applications by considering examples of time-dependent normalizations that are step functions. In this paper we shall apply Theorem 1.1 to obtain other results which involve time-dependent operators that are step functions.…”