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Transactions of the American Mathematical Society

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Extremal problems and symmetrization
for plane ring domains

Authors: A. Yu. Solynin and M. Vuorinen
Journal: Trans. Amer. Math. Soc. 348 (1996), 4095-4112
MSC (1991): Primary 30C85; Secondary 31A15
MathSciNet review: 1333399
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that Teichmüller's classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants.

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  • [A] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. MR 0200442
  • [B] Albert Baernstein II, Some topics in symmetrization, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 111–123. MR 1013818, 10.1007/BFb0086796
  • [Bat] Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman. MR 0066496
  • [D] V. N. Dubinin, Symmetrization in geometric theory of functions (Russian), Uspehi Mat. Nauk 49 (1994), 3-76. CMP 95:05
  • [F] J. Ferrand, Conformal capacities and extremal metrics, Manuscript, January 1994.
  • [FC] Henry E. Fettis and James C. Caslin, A table of the complete elliptic integral of the first kind for complex values of the modulus. Part I, ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, Wright-Patterson Air Force Base, Ohio, 1969. MR 0253511
  • [GS] F. P. Gardiner and D. P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683-736. CMP 92:16
  • [G] F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519. MR 0132841, 10.1090/S0002-9947-1961-0132841-2
  • [HLM] David A. Herron, Xiang Yang Liu, and David Minda, Ring domains with separating circles or separating annuli, J. Analyse Math. 53 (1989), 233–252. MR 1014988, 10.1007/BF02793416
  • [J1] J. Jenkins, Univalent functions and conformal mapping, Ergebnisse der Math. Vol. 18, Corrected ed., Springer-Verlag, Berlin--Heidelberg--New York, 1965.
  • [J2] J. Jenkins, On metrics defined by modules, Pacific J. Math. (to appear).
  • [K] G. V. Kuz′mina, Moduli of families of curves and quadratic differentials, Proc. Steklov Inst. Math. 1 (1982), vii+231. A translation of Trudy Mat. Inst. Steklov. 139 (1980). MR 664708
  • [LV] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
  • [S] Menahem Schiffer, On the modulus of doubly-connected domains, Quart. J. Math., Oxford Ser. 17 (1946), 197–213. MR 0018751
  • [SO1] A. Yu. Solynin, On the extremal decompositions of the plane or disk on two nonoverlapping domains (Russian), Kuban University, Krasnodar, 1984, Deponirovano in VINITI, N7800, 16p.
  • [SO2] A. Yu. Solynin, Moduli of doubly connected domains, and conformally invariant metrics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 196 (1991), no. Modul. Funktsii Kvadrat. Formy. 2, 122–131, 175 (Russian); English transl., J. Math. Sci. 70 (1994), no. 6, 2140–2146. MR 1164222, 10.1007/BF02111332
  • [SO3] A. Yu. Solynin, Solution of the Pólya-Szegő isoperimetric problem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsii. 9, 140–153, 190 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 311–320. MR 982489, 10.1007/BF01303655
  • [T] O. Teichmüller, Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math. 3 (1938), 621--678.
  • [TS] M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898
  • [V1] Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174
  • [V2] Matti Vuorinen, Conformally invariant extremal problems and quasiconformal maps, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 172, 501–514. MR 1188388, 10.1093/qmathj/43.4.501

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Additional Information

A. Yu. Solynin
Affiliation: Steklov Institute, Fontanka 27, St. Petersburg 191011, Russia
Email: FAX: 007-812-3105377

M. Vuorinen
Affiliation: University of Helsinki, FIN-00100 Helsinki, Finland
Email: FAX: 358-0-1913213

Received by editor(s): November 18, 1994
Received by editor(s) in revised form: May 2, 1995
Article copyright: © Copyright 1996 American Mathematical Society