Abstract
In this paper we present a geometric proof of the following fact: Let D be a Jordan domain in \(\mathbb {C}\), and let f be analytic on cl(D). Then there is an injective analytic map \(\phi :D\rightarrow \mathbb {C}\), and a polynomial p, such that \(f\equiv p\circ \phi \) on D (that is, f has a polynomial conformal model p).
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Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Two-dimensional shapes and lemniscates. Contemp. Math. 553, 45–59 (2011)
Hilbert, D.: Über die Entwicklung einer beliebigen analytischen Funktion einer Variablen in eine unendliche anch ganzen rationalen Funktionen fortschreitende Reihe. Göttinger Nachrichten, pp. 63–70, (1897)
Lowther, G., Speyer D.: Conjecture: every analytic function on the closed disk is conformally a polynomial. http://math.stackexchange.com/questions/437598/conjecture-every-analytic-function-on-the-closed-disk-is-conformally-a-polynomi (2013). Accessed: 22 June 2015
Pfluger, A.: Über die Konstruktion Riemannscher Flächen durch Verheftung. J. Indian Math. Soc. 24, 401–412 (1961)
Richards, T., Younsi, M.: Conformal models and fingerprints of pseudo-lemniscates. Submitted to Constructive Approximation (2015)
Richards, T.: Level curve configurations and conformal equivalence of meromorphic functions. Comput. Methods Funct. Theory 15(2), 323–371 (2015)
Younsi, M.: Shapes, fingerprints and rational lemniscates. Proc. Amer. Math. Soc. 144, 1087–1093 (2016)
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Communicated by Kenneth Stephenson.
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Richards, T. Conformal Equivalence of Analytic Functions on Compact Sets. Comput. Methods Funct. Theory 16, 585–608 (2016). https://doi.org/10.1007/s40315-016-0161-3
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DOI: https://doi.org/10.1007/s40315-016-0161-3