Runge pairs of Φ-like domains
DOI:
https://doi.org/10.24193/subbmath.2022.2.02Keywords:
Φ-like, Runge, semigroup, starlike, spirallike.Abstract
We prove that if E ⊆ Cn is a Φ-like domain and D ⊆ E is a Φ-like domain, then (D; E) is a Runge pair. Certain applications, examples and questions are also provided.
Mathematics Subject Classification (2010): 32E30, 30C45, 32H99.
Received 18 January 2022; Accepted 20 January 2022.
References
Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989.
Abate, M., The infinitesimal generators of semigroups of holomorphic maps, Ann. Mat. Pura Appl., 161(1992), 167-180.
Almer, B., Sur quelques probl´emes de la th`eorie des fonctions de deux variables complexes, Arkiv f¨or Mat. Ast. o. Fys., 17(1922), 1-70.
Arosio, L., Bracci, F., Wold, E.F., Solving the Loewner PDE in complete hyperbolic starlike domains of Cn, Adv. Math., 242(2013), 209-216.
Arosio, L., Bracci, F., Wold, E.F., Embedding univalent functions in filtering Loewner chains in higher dimensions, Proc. Amer. Math. Soc., 143(2015), 1627-1634.
Anders´en, E., Lempert, L., On the group of holomorphic automorphisms of Cn, Invent. Math., 110(1992), 371-388.
Bedford, E., Local and global envelopes of holomorphy of domains in C2, Trans. Amer. Math. Soc., 292(1985), 665-674.
Brickman, L., Φ-like analytic functions. I, Bull. Amer. Math. Soc., 79(1973), 555-558. [9] Chirka, E.M., Approximation by polynomials on star-shaped subsets of Cn, Math. Notes, 14(1972), 586-588.
Docquier, F., Grauert, H., Levisches problem und Rungescher Satz fu¨r Teilgebiete Stein- scher Mannigfaltigkeiten, Math. Ann., 140(1960), 94-123.
El Kasimi, A., Approximation polynoˆmiale dans les domaines ´etoil´es de Cn, Complex Var. Theory Appl., 10(1988), 179-182.
Fornaess, J.E., Zame, B., Runge exhaustions of domains in Cn, Math. Z., 194(1987), 1-5.
Graham, I., Hamada, H., Kohr, G., Kohr, M., Asymptotically spirallike mappings in several complex variables, J. Anal. Math., 105(2008), 267-302.
Graham, I., Hamada, H., Kohr, G., Kohr, M., Univalent subordination chains in reflexive complex Banach spaces, In: “Complex analysis and dynamical systems V”, Contemp. Math., 591, Amer. Math. Soc., Providence, RI, 2013, 83-111.
Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003.
Gurganus, K., Φ-like holomorphic functions in Cn and Banach spaces, Trans. Amer. Math. Soc., 205(1975), 389-406.
Hamada, H., Approximation properties on spirallike domains of Cn, Adv. Math., 268(2015), 467-477.
Ilyashenko, Y., Yakovenko, S., Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86, American Mathematical Society, Providence, 2008.
Jarnicki, M., Pflug, P., Extension of Holomorphic Functions, Walter de Gruyter, Berlin, 2000.
Kaup, L., Kaup, B., Holomorphic Functions of Several Variables, Walter de Gruyter, Berlin, 1983.
Laufer, H., Serre duality and envelopes of holomorphy, Trans. Amer. Math. Soc., 128(1966), 414-436.
Narasimhan, R., Analysis on Real and Complex Manifolds, Masson & Cie, Paris, 1973. [23] Ohsawa, T., Analysis of Several Complex Variables, Translations of Mathematical Monographs, 211, American Mathematical Society, 2002.
Range, M., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer, New York, 1986.
Reich, S., Shoikhet, D., Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.
Rudin, W., Real and Complex Analysis, 3rd edn, McGraw-Hill, New York, 1987.
Stout, E.L., Polynomial Convexity, Progress in Mathematics, 261, Birkha¨user, Boston, 2007.
Suffridge, T.J., Starlike and convex maps in Banach spaces, Pacific J. Math., 46(1973), 575-589.
Trapani, S., Complex retractions and envelopes of holomorphy, Proc. Amer. Math. Soc., 104(1988), 145-148.
Wermer, J., An example concerning polynomial convexity, Math. Ann., 139(1959), 147- 150. Addendum: Math. Ann., 140(1960), 322-323.
Wold, E.F., A Fatou-Bieberbach domain in C2 which is not Runge, Math. Ann., 340(2008), 775-780.
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