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Gennadiĭ Mikhaĭlovich Goluzin and geometric function theory. (English. Russian original) Zbl 1137.01022

St. Petersbg. Math. J. 18, No. 3, 347-372 (2007); translation from Algebra Anal. 18, No. 3, 3-38 (2006).
Gennadiĭ Mikhailovich Goluzin (1906–1952) worked mainly in the area of geometric function theory, understood as the study of general classes of functions defined on a domain in the plane or on a Riemannian surface, and his other major area was the theory of conformal mappings. This paper considers him as one of the classics of geometric function theory and presents his contributions to it, with an emphasis upon his monograph “The Geometric Theory of Functions of a Complex Variable” (Russian) (1952, translated into German (1957; Zbl 0083.06604) and into English (1969; Zbl 0183.07502)). The paper is completed with the list of 58 Goluzin’s research papers and another list of 62 quoted papers.

MSC:

01A70 Biographies, obituaries, personalia, bibliographies
30-03 History of functions of a complex variable
01A60 History of mathematics in the 20th century
30C55 General theory of univalent and multivalent functions of one complex variable
30C35 General theory of conformal mappings

Biographic References:

Goluzin, Gennadiĭ Mikhaĭlovich
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[1] Параметрические продолжения в теории однолистных функций., Издат. ”Наука”, Мосцощ, 1976 (Руссиан).
[2] Формулы Карлемана в комплексном анализе, ”Наука” Сибирск. Отдел., Новосибирск, 1990 (Руссиан). Первые приложения. [Фирст апплицатионс]. Лев Аизенберг, Царлеман’с формулас ин цомплеш аналысис, Матхематицс анд иц Апплицатионс, вол. 244, Клущер Ацадемиц Публишерс Гроуп, Дордречт, 1993. Тхеоры анд апплицатионс; Ревисед транслатион оф тхе 1990 Руссиан оригинал.
[3] Геометрическая теория функций комплексного переменного, Сецонд едитион. Едитед бы В. И. Смирнов. Щитх а супплемент бы Н. А. Лебедев, Г. В. Кузмина анд Ју. Е. Аленицын, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). Г. М. Голузин, Геометриц тхеоры оф фунцтионс оф а цомплеш вариабле, Транслатионс оф Матхематицал Монограпхс, Вол. 26, Америцан Матхематицал Социеты, Провиденце, Р.И., 1969. · Zbl 0148.30603
[4] G. P. Bakhtina, Variational methods and quadratic differentials in the problems on nonoverlapping domains, Thesis, Kiev, 1975, 12 pp. (Russian)
[5] 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. · Zbl 1057.30018 · doi:10.1512/iumj.2004.53.2211
[6] 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. · Zbl 1017.30033
[7] 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. · Zbl 1047.30016 · doi:10.2140/pjm.2003.212.13
[8] 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. · Zbl 0853.30021 · doi:10.1007/BF01057004
[9] L. de Branges, A proof of the Bieberbach conjecture, LOMI Preprints, no. E-5-84, Leningrad, 1984. (English)
[10] Louis de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137 – 152. · Zbl 0573.30014 · doi:10.1007/BF02392821
[11] V. N. Dubinin, Change of harmonic measure in symmetrization, Mat. Sb. (N.S.) 124(166) (1984), no. 2, 272 – 279 (Russian). · Zbl 0548.30017
[12] V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), no. 1, 49 – 55, 169 (Russian).
[13] 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. Funktsiĭ. 9, 48 – 66, 188 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 252 – 263. · Zbl 0717.30017 · doi:10.1007/BF01303649
[14] -, On the maximum of one conformal invariant, Preprint, Akad. Nauk SSSR Dal’nevost. Otdel., Inst. Prikl. Mat., Vladivostok, 1990. (Russian)
[15] 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. · doi:10.1070/RM1994v049n01ABEH002002
[16] 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.
[17] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. · Zbl 0514.30001
[18] E. G. Emel\(^{\prime}\)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.
[19] E. G. Emel\(^{\prime}\)yanov and G. V. Kuz\(^{\prime}\)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. · Zbl 0940.30018 · doi:10.1007/BF02172467
[20] 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. · Zbl 0483.30013
[21] 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). · Zbl 0552.30016
[22] Carl H. FitzGerald and Ch. Pommerenke, The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), no. 2, 683 – 690. · Zbl 0574.30018
[23] O. M. Fomenko and G. V. Kuz\(^{\prime}\)mina, The last 100 days of the Bieberbach conjecture, Math. Intelligencer 8 (1986), no. 1, 40 – 47. · Zbl 0616.30010 · doi:10.1007/BF03023920
[24] Геометрическая теория функций комплексного переменного, Сецонд едитион. Едитед бы В. И. Смирнов. Щитх а супплемент бы Н. А. Лебедев, Г. В. Кузмина анд Ју. Е. Аленицын, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). Г. М. Голусин, Геометрисче Функтионентхеорие, Хочсчулбüчер фüр Матхематик, Бд. 31, ВЕБ Деуцчер Верлаг дер Щиссенсчафтен, Берлин, 1957 (Герман). Г. М. Голузин, Геометриц тхеоры оф фунцтионс оф а цомплеш вариабле, Транслатионс оф Матхематицал Монограпхс, Вол. 26, Америцан Матхематицал Социеты, Провиденце, Р.И., 1969.
[25] Helmut Grunsky, Lectures on theory of functions in multiply connected domains, Vandenhoeck & Ruprecht, Göttingen, 1978. Studia Mathematica, Skript 4. · Zbl 0371.30015
[26] W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. · Zbl 0904.30001
[27] James A. Jenkins, On a problem of Gronwall, Ann. of Math. (2) 59 (1954), 490 – 504. · Zbl 0059.06104 · doi:10.2307/1969714
[28] James A. Jenkins, Some theorems on boundary distortion, Trans. Amer. Math. Soc. 81 (1956), 477 – 500. · Zbl 0071.07203
[29] James A. Jenkins, On the existence of certain general extremal metrics, Ann. of Math. (2) 66 (1957), 440 – 453. · Zbl 0082.06301 · doi:10.2307/1969901
[30] James A. Jenkins, On the existence of certain general extremal metrics. II, Tohoku Math. J. (2) 45 (1993), no. 2, 249 – 257. · Zbl 0780.30019 · doi:10.2748/tmj/1178225919
[31] 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. · Zbl 0083.29606
[32] James A. Jenkins, On certain geometrical problems associated with capacity, Math. Nachr. 39 (1969), 349 – 356. · Zbl 0172.09901 · doi:10.1002/mana.19690390412
[33] 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. · doi:10.1090/surv/021/08
[34] L. I. Kolbina, Some extremal problems in conformal mapping, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 865 – 868 (Russian). · Zbl 0049.17704
[35] L. I. Kolbina, Conformal mapping of the unit circle onto mutually nonoverlapping regions, Vestnik Leningrad. Univ. 10 (1955), no. 5, 37 – 43 (Russian).
[36] Квазиконформные отображения — новые методы и приложения, ”Наука” Сибирск. Отдел., Новосибирск, 1984 (Руссиан). Самуил Л. Крусчкал анд Реинер Кüхнау, Чуасиконформе Аббилдунген — неуе Метходен унд Анщендунген, Теубнер-Теште зур Матхематик [Теубнер Тешц ин Матхематицс], вол. 54, БСБ Б. Г. Теубнер Верлагсгеселлсчафт, Леипзиг, 1983 (Герман). Щитх Енглиш, Френч анд Руссиан суммариес. С. Л. Крушкал\(^{\приме}\) анд Р. Кюнау, Квазиконформные отображения — новые методы и приложения, ”Наука” Сибирск. Отдел., Новосибирск, 1984 (Руссиан). Самуил Л. Крусчкал анд Реинер Кüхнау, Чуасиконформе Аббилдунген — неуе Метходен унд Анщендунген, Теубнер-Теште зур Матхематик [Теубнер Тешц ин Матхематицс], вол. 54, БСБ Б. Г. Теубнер Верлагсгеселлсчафт, Леипзиг, 1983 (Герман). Щитх Енглиш, Френч анд Руссиан суммариес.
[37] 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. Funktsiĭ. 9, 85 – 97, 188 – 189 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 277 – 284. · Zbl 0717.30018 · doi:10.1007/BF01303651
[38] G. V. Kuz\(^{\prime}\)mina, Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov. 139 (1980), 241 (Russian). G. V. Kuz\(^{\prime}\)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). G. V. Kuz\(^{\prime}\)mina, Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov. 139 (1980), 241 (Russian). G. V. Kuz\(^{\prime}\)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).
[39] G. V. Kuz\(^{\prime}\)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.
[40] G. V. Kuz\(^{\prime}\)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. Funktsiĭ. 10, 72 – 95, 184 – 185 (Russian); English transl., J. Soviet Math. 59 (1992), no. 6, 1180 – 1196. · doi:10.1007/BF01374080
[41] G. V. Kuz\(^{\prime}\)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. G. V. Kuz\(^{\prime}\)mina, Erratum: ”Methods of the geometric theory of functions. I, II” [Algebra i Analiz 9 (1997), no. 3, 41 – 103; MR1466796 (98h:30041); ibid. 9 (1997), no. 5, 1 – 50; MR1604397 (99c:30047a)], Algebra i Analiz 10 (1998), no. 3, 223 (Russian); English transl., St. Petersburg Math. J. 10 (1999), no. 3, 577.
[42] G. V. Kuz\(^{\prime}\)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. · Zbl 1162.30011 · doi:10.1007/s10958-005-0320-y
[43] N. A. Lebedev, Majorizing region for the expression \?=\?\?(\?^{\?}\?’(\?)^{1-\?})/\?(\?)^{\?} in the class \?, Vestnik Leningrad. Univ. 10 (1955), no. 8, 29 – 41 (Russian).
[44] N. A. Lebedev, Some estimates for functions regular and univalent in a circle, Vestnik Leningrad. Univ. 10 (1955), no. 11, 3 – 21 (Russian).
[45] Принцип площадей в теории однолистных функций., Издат. ”Наука”, Мосцощ, 1975 (Руссиан).
[46] Y. J. Leung, On the \?th diameter problem in the class \Sigma , Complex Variables Theory Appl. 9 (1987), no. 2-3, 227 – 239. · Zbl 0636.30021
[47] Y. J. Leung and G. Schober, The \?th diameter problem in the class \Sigma , J. Analyse Math. 48 (1987), 247 – 266. · Zbl 0632.30027 · doi:10.1007/BF02790331
[48] Однолистные функции и ортонормированные системы, Издат. ”Наука”, Мосцощ, 1971 (Руссиан). И. М. Милин, Унивалент фунцтионс анд ортхонормал сыстемс, Америцан Матхематицал Социеты, Провиденце, Р. И., 1977. Транслатед фром тхе Руссиан; Транслатионс оф Матхематицал Монограпхс, Вол. 49.
[49] Jonathan R. Partington, Interpolation, identification, and sampling, London Mathematical Society Monographs. New Series, vol. 17, The Clarendon Press, Oxford University Press, New York, 1997. · Zbl 0892.93004
[50] Udo Pirl, Über die geometrische Gestalt eines Extremalkontinuums aus der Theorie der konformen Abbildung, Math. Nachr. 39 (1969), 297 – 312 (German). · Zbl 0202.07202 · doi:10.1002/mana.19690390408
[51] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. · Zbl 0044.38301
[52] Christian Pommerenke, Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159 – 173 (German). · Zbl 0184.30601 · doi:10.1515/crll.1965.218.159
[53] Ch. Pommerenke, On a variational method for univalent functions, Michigan Math. J. 17 (1970), 1 – 3. · Zbl 0175.36504
[54] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. · Zbl 0298.30014
[55] Edgar Reich and Menahem Schiffer, Estimates for the transfinite diameter of a continuum, Math. Z. 85 (1964), 91 – 106. · Zbl 0129.29304 · doi:10.1007/BF01114881
[56] 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. · Zbl 0049.06003
[57] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. · Zbl 0059.06901
[58] 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. · Zbl 0597.32021
[59] 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. Funktsiĭ. 9, 140 – 153, 190 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 311 – 320. · Zbl 0717.30020 · doi:10.1007/BF01303655
[60] 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.
[61] 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. · Zbl 1060.31001 · doi:10.4007/annals.2004.159.277
[62] Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. · Zbl 0547.30001
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