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Gennadii Mikhailovich Goluzin and geometric function theory


Author: G. V. Kuz'mina
Translated by: N. Yu. Netsvetaev
Original publication: Algebra i Analiz, tom 18 (2006), nomer 3.
Journal: St. Petersburg Math. J. 18 (2007), 347-372
MSC (2000): Primary 30-02, 30C55
Published electronically: April 10, 2007
MathSciNet review: 2255849
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Abstract | References | Similar Articles | Additional Information

Abstract: G. M. Goluzin crucially influenced the development and extension of geometric function theory. His results received world-wide recognition, and his monograph ``Geometric theory of functions of a complex variable'' has been a reference book for several generations of analysts.

This paper is a survey of Goluzin's scientific work on the occasion of the 100th anniversary of his birth.


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  • [A] I. A. \cyr{A}leksandrov, Parametricheskie prodolzheniya v teorii odnolistnykh funktsii, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0480952
  • [Ai] L. A. Aĭzenberg, Formuly Karlemana v kompleksnom analize, “Nauka” Sibirsk. Otdel., Novosibirsk, 1990 (Russian). Pervye prilozheniya. [First applications]. MR 1089612
    Lev Aizenberg, Carleman’s formulas in complex analysis, Mathematics and its Applications, vol. 244, Kluwer Academic Publishers Group, Dordrecht, 1993. Theory and applications; Revised translation of the 1990 Russian original. MR 1256735
  • [AlKuzL] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Second edition. Edited by V. I. Smirnov. With a supplement by N. A. Lebedev, G. V. Kuzmina and Ju. E. Alenicyn, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0219714
    G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [B] G. P. Bakhtina, Variational methods and quadratic differentials in the problems on nonoverlapping domains, Thesis, Kiev, 1975, 12 pp. (Russian)
  • [BaSo] Roger W. Barnard and Alexander Yu. Solynin, Local variations and minimal area problem for Carathéodory functions, Indiana Univ. Math. J. 53 (2004), no. 1, 135–167. MR 2048187, 10.1512/iumj.2004.53.2211
  • [BaPSo1] Roger W. Barnard, Kent Pearce, and Alexander Yu. Solynin, An isoperimetric inequality for logarithmic capacity, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 419–436. MR 1922198
  • [BaPSo2] Roger W. Barnard, Kent Pearce, and Alexander Yu. Solynin, Area, width, and logarithmic capacity of convex sets, Pacific J. Math. 212 (2003), no. 1, 13–23. MR 2016565, 10.2140/pjm.2003.212.13
  • [BarKh] V. P. Khavin and V. A. Bart, Szegő-Kolmogorov-Kreĭn theorems on a weighted trigonometric approximation, and Carleman-type formulas, Ukraïn. Mat. Zh. 46 (1994), no. 1-2, 100–127 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 46 (1994), no. 1-2, 101–132. MR 1294817, 10.1007/BF01057004
  • [Br1] L. de Branges, A proof of the Bieberbach conjecture, LOMI Preprints, no. E-5-84, Leningrad, 1984. (English)
  • [Br2] Louis de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137–152. MR 772434, 10.1007/BF02392821
  • [D1] V. N. Dubinin, Change of harmonic measure in symmetrization, Mat. Sb. (N.S.) 124(166) (1984), no. 2, 272–279 (Russian). MR 746071
  • [D3] V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), no. 1, 49–55, 169 (Russian). MR 804180
  • [D4] V. N. Dubinin, A separating transformation of domains, and problems on extremal decomposition, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsii. 9, 48–66, 188 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 252–263. MR 982483, 10.1007/BF01303649
  • [D5] -, On the maximum of one conformal invariant, Preprint, Akad. Nauk SSSR Dal'nevost. Otdel., Inst. Prikl. Mat., Vladivostok, 1990. (Russian)
  • [D6] V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3–76 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 1, 1–79. MR 1307130, 10.1070/RM1994v049n01ABEH002002
  • [D7] V. N. Dubinin, Conformal mappings and inequalities for algebraic polynomials, Algebra i Analiz 13 (2001), no. 5, 16–43 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 717–737. MR 1882862
  • [Du] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [E] E. G. Emel′yanov, Some properties of moduli of families of curves, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 144 (1985), 72–82, 175 (Russian). Analytic number theory and the theory of functions, 6. MR 787415
  • [EKuz] E. G. Emel′yanov and G. V. Kuz′mina, Theorems on extremal partitioning in a family of systems of domains of different types, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 74–104, 229 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2221–2239. MR 1691285, 10.1007/BF02172467
  • [F1] S. I. Fedorov, On the maximum of a conformal invariant in a problem on nonoverlapping domains, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 112 (1981), 172–183, 202 (Russian). Analytic number theory and the theory of functions, 4. MR 644003
  • [F2] S. I. Fedorov, Chebotarev’s variational problem in the theory of the capacity of plane sets, and covering theorems for univalent conformal mappings, Mat. Sb. (N.S.) 124(166) (1984), no. 1, 121–139 (Russian). MR 743060
  • [FiPom] Carl H. FitzGerald and Ch. Pommerenke, The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), no. 2, 683–690. MR 792819, 10.1090/S0002-9947-1985-0792819-9
  • [FoKuz] O. M. Fomenko and G. V. Kuz′mina, The last 100 days of the Bieberbach conjecture, Math. Intelligencer 8 (1986), no. 1, 40–47. MR 823219, 10.1007/BF03023920
  • [G] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Second edition. Edited by V. I. Smirnov. With a supplement by N. A. Lebedev, G. V. Kuzmina and Ju. E. Alenicyn, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0219714
    G. M. Golusin, Geometrische Funktionentheorie, Hochschulbücher für Mathematik, Bd. 31, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957 (German). MR 0089896
    G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [Gr] Helmut Grunsky, Lectures on theory of functions in multiply connected domains, Vandenhoeck & Ruprecht, Göttingen, 1978. Studia Mathematica, Skript 4. MR 0463413
  • [H] W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. MR 1310776
  • [J1] James A. Jenkins, On a problem of Gronwall, Ann. of Math. (2) 59 (1954), 490–504. MR 0061170
  • [J2] James A. Jenkins, Some theorems on boundary distortion, Trans. Amer. Math. Soc. 81 (1956), 477–500. MR 0076862, 10.1090/S0002-9947-1956-0076862-5
  • [J3-1] James A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. (2) 66 (1957), 440–453. MR 0090648
  • [J3-2] James A. Jenkins, On the existence of certain general extremal metrics. II, Tohoku Math. J. (2) 45 (1993), no. 2, 249–257. MR 1215927, 10.2748/tmj/1178225919
  • [J4] James A. Jenkins, Univalent functions and conformal mapping, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
  • [J5] James A. Jenkins, On certain geometrical problems associated with capacity, Math. Nachr. 39 (1969), 349–356. MR 0249594
  • [J6] James A. Jenkins, The method of the extremal metric, The Bieberbach conjecture (West Lafayette, Ind., 1985) Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 95–104. MR 875234, 10.1090/surv/021/08
  • [K1] L. I. Kolbina, Some extremal problems in conformal mapping, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 865–868 (Russian). MR 0048582
  • [K2] L. I. Kolbina, Conformal mapping of the unit circle onto mutually nonoverlapping regions, Vestnik Leningrad. Univ. 10 (1955), no. 5, 37–43 (Russian). MR 0070723
  • [KrKuh] Samuil L. Kruschkal and Reiner Kühnau, Quasikonforme Abbildungen—neue Methoden und Anwendungen, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 54, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1983 (German). With English, French and Russian summaries. MR 730760
    Samuil L. Kruschkal and Reiner Kühnau, Quasikonforme Abbildungen—neue Methoden und Anwendungen, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 54, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1983 (German). With English, French and Russian summaries. MR 730760
  • [Ku] V. O. Kuznetsov, Properties of associated quadratic differentials in some extremal problems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsii. 9, 85–97, 188–189 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 277–284. MR 982485, 10.1007/BF01303651
  • [Kuz1] G. V. Kuz′mina, Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov. 139 (1980), 241 (Russian). MR 612632
    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
  • [Kuz2] G. V. Kuz′mina, On the problem of the maximum of the product of conformal radii of nonoverlapping domains, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 100 (1980), 131–145, 175–176 (Russian). Analytic number theory and the theory of functions, 3. MR 599943
  • [Kuz3] G. V. Kuz′mina, On the extremal partitioning of the Riemann sphere, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990), no. Anal. Teor. Chisel i Teor. Funktsii. 10, 72–95, 184–185 (Russian); English transl., J. Soviet Math. 59 (1992), no. 6, 1180–1196. MR 1097590, 10.1007/BF01374080
  • [Kuz4] G. V. Kuz′mina, Methods of the geometric theory of functions. II, Algebra i Analiz 9 (1997), no. 5, 1–50 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 5, 889–930. MR 1604397
  • [Kuz5] G. V. Kuz′mina, The extremal metric method in problems of maximizing the product of powers of conformal radii of nonoverlapping domains in the presence of free parameters, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), no. Anal. Teor. Chisel i Teor. Funkts. 19, 52–67, 199 (Russian, with Russian summary); English transl., J. Math. Sci. (N. Y.) 129 (2005), no. 3, 3843–3851. MR 2023032, 10.1007/s10958-005-0320-y
  • [L1] N. A. Lebedev, Majorizing region for the expression 𝐼=𝑙𝑛(𝑧^{𝜆}𝑓’(𝑧)^{1-𝜆})/𝑓(𝑧)^{𝜆} in the class 𝑆, Vestnik Leningrad. Univ. 10 (1955), no. 8, 29–41 (Russian). MR 0072213
  • [L2] N. A. Lebedev, Some estimates for functions regular and univalent in a circle, Vestnik Leningrad. Univ. 10 (1955), no. 11, 3–21 (Russian). MR 0074514
  • [L3] N. A. \cyr{L}ebedev, Printsip ploshchadei v teorii odnolistnykh funktsii, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0450540
  • [Le] Y. J. Leung, On the 𝑁th diameter problem in the class Σ, Complex Variables Theory Appl. 9 (1987), no. 2-3, 227–239. MR 923223
  • [LeScho] Y. J. Leung and G. Schober, The 𝑁th diameter problem in the class Σ, J. Analyse Math. 48 (1987), 247–266. MR 910011, 10.1007/BF02790331
  • [M] I. M. Milin, Odnolistnye funktsii i ortonormirovannye sistemy, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0369684
    I. M. Milin, Univalent functions and orthonormal systems, American Mathematical Society, Providence, R. I., 1977. Translated from the Russian; Translations of Mathematical Monographs, Vol. 49. MR 0427620
  • [P] Jonathan R. Partington, Interpolation, identification, and sampling, London Mathematical Society Monographs. New Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1997. MR 1473224
  • [Pi] Udo Pirl, Über die geometrische Gestalt eines Extremalkontinuums aus der Theorie der konformen Abbildung, Math. Nachr. 39 (1969), 297–312 (German). MR 0254223
  • [PoSz] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
  • [Pom1] Christian Pommerenke, Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159–173 (German). MR 0180669
  • [Pom2] Ch. Pommerenke, On a variational method for univalent functions, Michigan Math. J. 17 (1970), 1–3. MR 0255792
  • [Pom3] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
  • [RS] Edgar Reich and Menahem Schiffer, Estimates for the transfinite diameter of a continuum, Math. Z. 85 (1964), 91–106. MR 0174721
  • [SchaSp] A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht Functions, American Mathematical Society Colloquium Publications, Vol. 35, American Mathematical Society, New York, N. Y., 1950. With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad. MR 0037908
  • [SSp] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. MR 0065652
  • [So1] A. Yu. Solynin, The dependence of the problem of moduli for a family of some classes of curves on parameters, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 144 (1985), 136–145, 177 (Russian). Analytic number theory and the theory of functions, 6. MR 787420
  • [So2] 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
  • [So3] A. Yu. Solynin, Moduli and extremal metric problems, Algebra i Analiz 11 (1999), no. 1, 3–86 (Russian); English transl., St. Petersburg Math. J. 11 (2000), no. 1, 1–65. MR 1691080
  • [SoZ] Alexander Yu. Solynin and Victor A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. of Math. (2) 159 (2004), no. 1, 277–303. MR 2052355, 10.4007/annals.2004.159.277
  • [St] Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423

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Additional Information

G. V. Kuz'mina
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: kuzmina@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-07-00954-5
Keywords: Univalent function, conformal mapping, geometric function theory, quadratic differential, method of the extremal metric, symmetrization method
Received by editor(s): March 30, 2006
Published electronically: April 10, 2007
Dedicated: Dedicated to the 100th anniversary of Gennadiĭ Mikhaĭlovich Goluzin’s birth
Article copyright: © Copyright 2007 American Mathematical Society