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Parametric Representation of Univalent Functions with Boundary Regular Fixed Points

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Abstract

A classical result in the theory of Loewner’s parametric representation states that the semigroup \({{\mathfrak {U}}}_0\) of all conformal self-maps \(\varphi \) of the unit disk \({\mathbb {D}}\) normalized by \(\varphi (0) = 0\) and \(\varphi '(0) > 0\) can be obtained as the reachable set of the Loewner–Kufarev control system

$$\begin{aligned} \frac{\mathrm {d}w_t}{\mathrm {d}t}=G_t\circ w_t,\quad t\geqslant 0,\qquad w_0=\mathsf{id}_{\mathbb {D}}, \end{aligned}$$

where the control functions \(t\mapsto G_t\in \mathsf{Hol}({\mathbb {D}},{\mathbb {C}})\) form a convex cone. We extend this result to semigroups \({{\mathfrak {U}}}[F]\) formed by all conformal self-maps of \({\mathbb {D}}\) with the prescribed finite set F of boundary regular fixed points and to their counterparts \({{\mathfrak {U}}}_{\tau }[F]\) for the case of self-maps having the Denjoy–Wolff point at \(\tau \in {\overline{{\mathbb {D}}}}{\setminus } F\).

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Notes

  1. Loewner [33] himself obtained the parametric representation of a dense subclass of \(\mathcal {S}\). Later it was extended to the whole class by Pommerenke [38, 39] and independently by Gutljanskiĭ [30].

  2. Observe that our notation \(\Lambda \) coincides with the spectral function in [12] taken with the opposite sign.

  3. Of course, in the general case, some additional condition of topological nature would also be required.

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Acknowledgements

Much inspiration for the work under the present paper has been drawn from the scientific publications of and from the personal communication with Prof. Viktor V. Goryainov. The author is also grateful to Prof. Oliver Roth for pointing out reference [42] essentially used in the proof. The text of the paper has been considerably improved thanks to valuable suggestions of the anonymous referees.

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  1. Department of Mathematics and Natural Sciences, University of Stavanger, 4036, Stavanger, Norway

    Pavel Gumenyuk

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Communicated by Stephan Ruscheweyh.

Partially supported by the FIRB grant Futuro in Ricerca “Geometria Differenziale Complessa e Dinamica Olomorfa” n. RBFR08B2HY and by Ministerio de Economía y Competitividad (Spain) project MTM2015-63699-P.

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Gumenyuk, P. Parametric Representation of Univalent Functions with Boundary Regular Fixed Points. Constr Approx 46, 435–458 (2017). https://doi.org/10.1007/s00365-017-9376-4

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