St. Petersburg Mathematical Society

Author(s): 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
Posted: April 10, 2007
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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.

[A]: I. A. Aleksandrov, Parametric continuations in the theory of univalent functions, ``Nauka'', Moscow, 1976. (Russian) MR 0480952 (58:1099)
[Ai]: L. A. Aizenberg, Carleman's formulas in complex analysis, ``Nauka'', Novosibirsk, 1990; English transl., Math. Appl., vol. 244, Kluwer Acad. Publ. Group, Dordrecht, 1993. MR 1089612 (92b:32007); MR 1256735 (95d:32004)
[AlKuzL]: Yu. E. Alenitsin, G. V. Kuz'mina, and N. A. Lebedev, Methods and results in geometric function theory, G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, 2nd ed., ``Nauka'', Moscow, 1966, pp. 532-626; English transl., Transl. Math. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, 1969. MR 0219714 (36:2793); MR 0247039 (40:308)
[B]: G. P. Bakhtina, Variational methods and quadratic differentials in the problems on nonoverlapping domains, Thesis, Kiev, 1975, 12 pp. (Russian)
[BaSo]: R. W. Barnard and A. Yu. Solynin, Local variations and minimal area problem for Carathéodory functions, Indiana Univ. Math. J. 53 (2004), no. 1, 135-167. MR 2048187 (2005a:30028)
[BaPSo1]: R. W. Barnard, K. Pearce, and A. Yu. Solynin, An isoperimetric inequality for logarithmic capacity, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 419-436. MR 1922198 (2003g:30039)
[BaPSo2]: -, Area, width, and logarithmic capacity of convex sets, Pacific J. Math. 212 (2003), no. 1, 13-23. MR 2016565 (2004j:30041)
[BarKh]: V. A. Bart and V. P. Khavin, Szegö-Kolmogorov-Krein theorems on a weighted trigonometric approximation, and Carleman-type formulas, Ukrain. Mat. Zh. 46 (1994), no. 1-2, 100-127; English transl., Ukrainian Math. J. 46 (1994), no. 1-2, 101-132. MR 1294817 (95k:42004)
[Br1]: L. de Branges, A proof of the Bieberbach conjecture, LOMI Preprints, no. E-5-84, Leningrad, 1984. (English)
[Br2]: -, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137-152. MR 772434 (86h:30026)
[D1]: V. N. Dubinin, Change of harmonic measure in symmetrization, Mat. Sb. (N.S.) 124 (1984), no. 2, 272-279; English transl. in Math. USSR-Sb. 52 (1985). MR 746071 (85j:30049)
[D3]: -, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), no. 1, 49-55; English transl., Math. Notes 38 (1985), no. 1-2, 539-542. MR 804180 (87j:31005)
[D4]: -, A separating transformation of domains, and problems on extremal decomposition, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), 48-66; English transl., J. Soviet Math. 53 (1991), no. 3, 252-263. MR 982483 (90c:30006)
[D5]: -, On the maximum of one conformal invariant, Preprint, Akad. Nauk SSSR Dal'nevost. Otdel., Inst. Prikl. Mat., Vladivostok, 1990. (Russian)
[D6]: -, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1, 3-76; English transl., Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054)
[D7]: -, Conformal mappings and inequalities for algebraic polynomials, Algebra i Analiz 13 (2001), no. 5, 16-43; English transl., St. Petersburg Math. J. 13 (2002), no. 5, 717-737. MR 1882862 (2003j:30010)
[Du]: P. Duren, Univalent functions, Grundlehren Math. Wiss., vol. 259, Springer-Verlag, New York-Berlin, 1983. MR 708494 (85j:30034)
[E]: E. G. Emel'yanov, Some properties of the moduli of families of curves, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 144 (1985), 72-82; English transl., J. Soviet Math. 38 (1987), no. 4, 2081-2090. MR 787415 (87i:30040)
[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), 74-104; English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2221-2239. MR 1691285 (2000c:30082)
[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; English transl. in J. Soviet Math. 25 (1984), no. 2. MR 644003 (83e:30026)
[F2]: -, 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 (1984), no. 1, 121-139; English transl. in Math. USSR-Sb. 52 (1985), no. 1. MR 743060 (85j:30006)
[FiPom]: C. H. FitzGerald and Ch. Pommerenke, The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), no. 2, 683-690. MR 792819 (87b:30023)
[FoKuz]: O. M. Fomenko and G. V. Kuz'mina, The last days of the Bieberbach conjecture, Math. Intelligencer 8 (1986), no. 1, 40-47. MR 823219 (87h:01059)
[G]: G. M. Goluzin, Geometrical theory of functions of a complex variable, ``GITTL'', Moscow-Leningrad, 1952; 2nd ed., ``Nauka'', Moscow, 1966; German transl. of 1st ed., VEB Deutscher Verlag Wiss., Berlin, 1957; English transl. of 2nd ed., Transl. Math. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, 1969. MR 0219714 (36:2793); MR 0219714 (36:2793); MR 0089896 (19:735e); MR 0247039 (40:308)
[Gr]: H. Grunsky, Lectures on theory of functions in multiply connected domains, Vandenhoeck and Ruprecht, Göttingen, 1978. MR 0463413 (57:3365)
[H]: W. K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., No. 48, Cambridge Univ. Press, Cambridge, 1958; 2nd ed., Cambridge Tracts in Math., No. 110, 1994. MR 1310776 (96f:30003); MR 1310776 (96f:30003)
[J1]: J. A. Jenkins, On a problem of Gronwall, Ann. of Math. (2) 59 (1954), no. 3, 490-504. MR 0061170 (15:786b)
[J2]: -, Some theorems on boundary distortion, Trans. Amer. Math. Soc. 81 (1956), no. 2, 477-500. MR 0076862 (17:956b)
[J3-1]: -, On the existence of certain general extremal metrics, Ann. of Math. (2) 66 (1957), 440-453. MR 0090648 (19:845g)
[J3-2]: -, On the existence of certain general extremal metrics. II, Tohoku Math. J. (2) 45 (1993), no. 2, 249-257. MR 1215927 (94h:31002)
[J4]: -, Univalent functions and conformal mappings, Ergeb. Math. Grenzgeb. (N. F.), Bd. 18, Springer-Verlag, Berlin, 1958; 2nd ed. corrected, 1965. MR 0096806 (20:3288)
[J5]: -, On certain geometrical problems associated with capacity, Math. Nachr. 39 (1969), no. 4-6, 349-356. MR 0249594 (40:2837)
[J6]: -, The method of 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 (87k:30036)
[K1]: L. I. Kolbina, Some extremal problems in conformal mapping, Dokl. Akad. Nauk SSSR (N.S.) 84 (1952), no. 5, 865-868. (Russian) MR 0048582 (14:35e)
[K2]: -, Conformal mapping of the unit circle onto mutually nonoverlapping regions, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Khim. 1955, vyp. 2, 37-43. (Russian) MR 0070723 (17:26d)
[KrKuh]: S. L. Krushkal' and R. Kyunau, Quasiconformal mappings--new methods and applications, ``Nauka'', Novosibirsk, 1984; German transl., Teubner Texts in Math., vol. 54, Teubner, Leipzig, 1983. MR 730760 (85k:30032b); MR 0730760 (85k:30032b)
[Ku]: V. O. Kuznetsov, Properties of associated quadratic differentials in some extremal problems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), 85-97; English transl., J. Soviet Math. 53 (1991), no. 3, 277-284. MR 982485 (90h:30057)
[Kuz1]: G. V. Kuz'mina, Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov. 139 (1980), 241 pp.; English transl., Proc. Steklov Inst. Math. 1982, no. 1, 231 pp. MR 612632 (84j:30038a); MR 0664708 (84j:30038b)
[Kuz2]: -, 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; English transl. in J. Soviet Math. 19 (1982), no. 6. MR 599943 (82c:30033)
[Kuz3]: -, On the extremal partitioning of the Riemann sphere, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990), 72-95; English transl., J. Soviet Math. 59 (1992), no. 6, 1180-1196. MR 1097590 (92k:30010)
[Kuz4]: -, Methods of geometric function theory. I, II, Algebra i Analiz 9 (1997), no. 3, 41-103; no. 5, 1-50; English transl., St. Petersburg Math. J. 9 (1998), no. 3, 455-507; no. 5, 889-930. MR 1604397 (99c:30047a); MR 1604397 (99c:30047a)
[Kuz5]: -, 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), 52-67; English transl., J. Math. Sci. (New York) 129 (2005), no. 3, 3843-3851. MR 2023032 (2004m:30034)
[L1]: N. A. Lebedev, Majorizing region for the expression $I=\ln z^{\la } f'(z)^{1-\lambda} f(z)^{-\lambda}$ in the class , Vestnik Leningrad. Univ. Ser. Mat. Fiz. Khim. 1955, vyp. 3, 29-41. (Russian) MR 0072213 (17:248b)
[L2]: -, Some estimates for functions regular and univalent in a circle, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Khim. 1955, vyp. 4, 3-21. (Russian) MR 0074514 (17:599c)
[L3]: -, The area principle in the theory of univalent functions, ``Nauka'', Moscow, 1975. (Russian) MR 0450540 (56:8834)
[Le]: Y. J. Leung, On the Nth diameter problem in the class $\Sigma$ , Complex Variables Theory Appl. 9 (1987), no. 2-3, 227-239. MR 923223 (89d:30011)
[LeScho]: Y. J. Leung and G. Schober, The th diameter problem in the class $\Sigma$ , J. Analyse Math. 48 (1987), 247-266. MR 910011 (89d:30012)
[M]: I. M. Milin, Univalent functions and orthonormal systems, ``Nauka'', Moscow, 1971; English transl., Transl. Math. Monogr., vol. 49, Amer. Math. Soc., Providence, RI, 1977. MR 0369684 (51:5916); MR 0427620 (55:651)
[P]: J. R. Partington, Interpolation, identification, and sampling, London Math. Soc. Monogr. (N.S.), vol. 17, The Clarendon Press, Oxford Univ. Press, New York, 1997. MR 1473224 (99m:41001)
[Pi]: U. Pirl, Über die geometrische Gestalt eines Extremalkontinuums aus der Theorie der konformen Abbildung, Math. Nachr. 39 (1969), no. 1-6, 297-312. MR 0254223 (40:7432)
[PoSz]: G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., No. 27, Princeton Univ. Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
[Pom1]: Ch. Pommerenke, Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159-173. MR 0180669 (31:4900)
[Pom2]: -, On a variational method for univalent functions, Michigan Math. J. 17 (1970), no. 1, 1-3. MR 0255792 (41:452)
[Pom3]: -, Univalent functions. With a chapter on quadratic differentials by G. Jensen, Math. Lehrbucher, Bd. 25, Vandenhoeck and Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
[RS]: E. Reich and M. Schiffer, Estimates for the transfinite diameter of a continuum, Math. Z. 85 (1964), no. 1, 91-106. MR 0174721 (30:4921)
[SchaSp]: A. C. Schaeffer and D. C. Spencer, Coefficient regions for schlicht functions, Amer. Math. Soc. Colloq. Publ., vol. 35, Amer. Math. Soc., New York, 1950. MR 0037908 (12:326c)
[SSp]: M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, NJ, 1954. MR 0065652 (16:461g)
[So1]: A. Yu. Solynin, The dependence on parameters of the modulus problem for families of several classes of curves, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 144 (1985), 136-145; English transl., J. Soviet Math. 38 (1987), no. 4, 2131-2139. MR 787420 (86h:30042)
[So2]: -, Solution of the Pólya-Szegö isoperimetric problem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), 140-153; English transl., J. Soviet Math. 53 (1991), no. 3, 311-320. MR 982489 (90h:30059)
[So3]: -, Moduli and extremal metric problems, Algebra i Analiz 11 (1999), no. 1, 3-86; English transl., St. Petersburg Math. J. 11 (2000), no. 1, 1-65. MR 1691080 (2001b:30058)
[SoZ]: A. Yu. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. of Math. (2) 159 (2004), no. 1, 277-303. MR 2052355 (2005a:31002)
[St]: K. Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. (3), Bd. 5, Springer-Verlag, Berlin-New York, 1984. MR 0743423 (86a:30072)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 30-02, 30C55

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: 10.1090/S1061-0022-07-00954-5
PII: S 1061-0022(07)00954-5
Keywords: Univalent function, conformal mapping, geometric function theory, quadratic differential, method of the extremal metric, symmetrization method
Received by editor(s): 30/MAR/2006
Posted: April 10, 2007
Dedicated: Dedicated to the 100th anniversary of Gennadii Mikhailovich Goluzin's birth
Copyright of article: Copyright 2007, American Mathematical Society

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ISSN: 1547-7371(e) ISSN: 1061-0022(p)

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