A classical result in the theory of Loewner’s parametric representation states that the semigroup U0 of all conformal self-maps φ of the unit disk D normalized by φ(0) = 0 and φ′(0) > 0 can be obtained as the reachable set of the Loewner–Kufarev control system dwt/dt Gt∘wt,t⩾0,w0=idD,where the control functions t↦ Gt∈ Hol(D, C) form a convex cone. We extend this result to semigroups U[F] formed by all conformal self-maps of D with the prescribed finite set F of boundary regular fixed points and to their counterparts Uτ[ F] for the case of self-maps having the Denjoy–Wolff point at τ∈ D¯ F.
Parametric Representation of Univalent Functions with Boundary Regular Fixed Points
Gumenyuk P.
2017-01-01
Abstract
A classical result in the theory of Loewner’s parametric representation states that the semigroup U0 of all conformal self-maps φ of the unit disk D normalized by φ(0) = 0 and φ′(0) > 0 can be obtained as the reachable set of the Loewner–Kufarev control system dwt/dt Gt∘wt,t⩾0,w0=idD,where the control functions t↦ Gt∈ Hol(D, C) form a convex cone. We extend this result to semigroups U[F] formed by all conformal self-maps of D with the prescribed finite set F of boundary regular fixed points and to their counterparts Uτ[ F] for the case of self-maps having the Denjoy–Wolff point at τ∈ D¯ F.File in questo prodotto:
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