Geodesic
Gennadii Mikhailovich Goluzin and geometric function theory
Algebra i analiz, Tome 18 (2006) no. 3, pp. 3-38.

Voir la notice de l'article provenant de la source Math-Net.Ru

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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G. V. Kuz'mina. Gennadii Mikhailovich Goluzin and geometric function theory. Algebra i analiz, Tome 18 (2006) no. 3, pp. 3-38. http://geodesic.mathdoc.fr/item/AA_2006_18_3_a0/

[1929] “O nekotorykh otsenkakh, otnosyaschikhsya k funktsiyam, sovershayuschim odnolistnoe konformnoe preobrazovanie kruga”, Mat. sb., 36:1/2 (1929), 152–172 | Zbl

[1933] “Obobschenie formuly Karlemana i prilozhenie ee k analiticheskomu prodolzheniyu funktsii”, Mat. sb., 40:2 (1933), 144–149 (with V. I. Krylovym) | Zbl

[1934] “Reshenie osnovnykh ploskikh zadach matematicheskoi fiziki dlya sluchaya uravneniya Laplasa i mnogosvyaznykh oblastei, ogranichennykh okruzhnostyami (metod funktsionalnykh uravnenii)”, Mat. sb., 41:2 (1934), 246–276 | Zbl

[1934a] “Reshenie prostranstvennoi zadachi Dirikhle dlya uravneniya Laplasa i dlya oblastei, ogranichennykh konechnym chislom sfer”, Mat. sb., 41:2 (1934), 277–283 | Zbl

[1935] “K teorii odnolistnykh konformnykh preobrazovanii”, Mat. sb., 42:2 (1935), 169–190 | Zbl

[1935a] “Reshenie ploskoi zadachi teploprovodnosti dlya mnogosvyaznykh oblastei, ogranichennykh okruzhnostyami, v sluchae nalichiya izoliruyuschego sloya”, Mat. sb., 42:2 (1935), 191–198 | Zbl

[1935b] “K printsipu mazhoratsii v teorii funktsii”, Mat. sb., 42:6 (1935), 647–650 | Zbl

[1936] “O teoremakh iskazheniya v teorii konformnykh otobrazhenii”, Mat. sb. (Nov. ser.), 1(43):1 (1936), 127–135 | Zbl

[1936a] “Sur la représentation conforme”, Mat. sb. (Nov. ser.), 1(43):3 (1936), 273–282 | Zbl

[1936b] “Sur les théorèmes de rotation dans la théorie des fonctions univalentes”, Mat. sb. (Nov. ser.), 1(43):3 (1936), 293–296 | Zbl

[1937] “O teoremakh iskazheniya dlya konformnogo otobrazheniya mnogosvyaznykh oblastei”, Mat. sb. (Nov. ser.), 2(44):1 (1937), 37–64 | Zbl

[1937a] “Nekotorye teoremy pokrytiya dlya funktsii, regulyarnykh v kruge”, Mat. sb. (Nov. ser.), 2(44):3 (1937), 617–619 | Zbl

[1937b] “Dopolnenie k rabote «O teoremakh iskazheniya v teorii konformnykh otobrazhenii»”, Mat. sb. (Nov. ser.), 2(44):4 (1937), 685–688 | Zbl

[1937c] “O konformnom otobrazhenii dvukhsvyaznykh oblastei, ogranichennykh pryamolineinymi i krugovymi mnogougolnikami”, Konformnoe otobrazhenie odnosvyaznykh i mnogosvyaznykh oblastei, ONTI, M.–L., 1937, 90–97

[1937d] “Konformnoe otobrazhenie mnogosvyaznykh oblastei na ploskost s razrezami metodom funktsionalnykh uravnenii”, Konformnoe otobrazhenie odnosvyaznykh i mnogosvyaznykh oblastei, ONTI, M.–L., 1937, 98–110

[1938] “Nekotorye otsenki koeffitsientov odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 3(45):2 (1938), 321–330 | Zbl

[1938a] “O metode nepreryvnosti v teorii konformnykh otobrazhenii mnogosvyaznykh oblastei”, Mat. sb. (Nov. ser.), 4(46):1 (1938), 3–8

[1939] “Iteratsionnye protsessy dlya konformnogo otobrazheniya mnogosvyaznykh oblastei”, Mat. sb. (Nov. ser.), 6(48):3 (1939), 377–382 | Zbl

[1939a] “O predelnykh znacheniyakh integrala Koshi”, Uchen. zap. LGU, 1939, no. 37, 43–47 | Zbl

[1939b] “O polnykh sistemakh funktsii v kompleksnoi oblasti”, Uchen. zap. LGU, 1939, no. 37, 48–51

[1939b] “K teorii odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 6(48):3 (1939), 383–388 | Zbl

[1939c] “Vnutrennie zadachi teorii odnolistnykh funktsii”, Uspekhi mat. nauk, 1939, no. 6, 26–89

[1940] “O $p$-listnykh funktsiyakh”, Mat. sb. (Nov. ser.), 8(50):2 (1940), 277–284 | MR | Zbl

[1943] “O koeffitsientakh odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 12(54):1 (1943), 40–47 | Zbl

[1943a] “K teorii odnolistnykh funktsii”, Mat. sb., 12(54):1 (1943), 48–55 | Zbl

[1943b] “K teorii kryla v ploskoparallelnom potoke zhidkosti”, Mat. sb. (Nov. ser.), 12(54):1 (1943), 146–151 | Zbl

[1945] “Otsenka proizvodnoi dlya funktsii, regulyarnykh i ogranichennykh v kruge”, Mat. sb. (Nov. ser.), 16(58):3 (1945), 295–306 | Zbl

[1946] “K teorii odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 18(60):1 (1946), 167–179 | Zbl

[1946a] “O zadache Karateodori–Feiera i ob odnoi analogichnoi zadache”, Mat. sb. (Nov. ser.), 18(60):2 (1946), 213–226 | Zbl

[1946b] “Nekotorye svoistva polinomov”, Mat. sb. (Nov. ser.), 18(60):2 (1946), 227–236 | Zbl

[1946c] “O teoremakh iskazheniya v teorii konformnykh otobrazhenii”, Mat. sb. (Nov. ser.), 18(60):3 (1946), 379–390 | Zbl

[1946d] “O chisle konechnykh asimptoticheskikh znachenii tselykh funktsii konechnogo poryadka”, Mat. sb. (Nov. ser.), 18(60):3 (1946), 391–396 | Zbl

[1946e] “O teoremakh iskazheniya i koeffitsientakh odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 19(61):2 (1946), 183–202 | Zbl

[1946f] “Metod variatsii v konformnom otobrazhenii. I”, Mat sb. (Nov. ser.), 19(61):2 (1946), 203–236 | Zbl

[1946g] “Otsenki dlya analiticheskikh funktsii s ogranichennym srednim modulem”, Tr. Mat. in-ta AN SSSR, 18, 1946, 3–87 | MR | Zbl

[1946h] “Metod variatsii v konformnom otobrazhenii”, Nauch. byull. LGU, 1946, no. 9, 3–5

[1947] “Metod variatsii v konformnom otobrazhenii. II”, Mat. sb. (Nov. ser.), 21(63):1 (1947), 83–117 | Zbl

[1947a] “Metod variatsii v konformnom otobrazhenii. III”, Mat. sb. (Nov. ser.), 21(63):1 (1947), 119–132 | Zbl

[1948] “Nekotorye teoremy pokrytiya v teorii analiticheskikh funktsii”, Mat. sb. (Nov. ser.), 22(64):3 (1948), 353–372 | Zbl

[1948a] “O koeffitsientakh odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 22(64):3 (1948), 373–380 | Zbl

[1948b] “O teoremakh iskazheniya i koeffitsientakh odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 23(65):3 (1948), 353–360 | Zbl

[1949] “Ob analiticheskikh funktsiyakh s ogranichennym srednim modulem”, Uchen. zap. LGU, 1949, no. 111, 120–125

[1949a] “Variatsionnyi vyvod teorem iskazheniya dlya odnolistnykh funktsii”, Uchen. zap. LGU, 1949, no. 111, 126–134

[1949b] “Nekotorye neravenstva dlya analiticheskikh funktsii”, Izv. AN Kazakhsk. SSR. Ser. mat. i mekh., 3 (1949), 101–105

[1949c] “Nekotorye voprosy teorii odnolistnykh funktsii”, Tr. Mat. in-ta AN SSSR, 27, 1949, 3–109 | MR | Zbl

[1949d] “O srednikh velichinakh”, Mat. sb. (Nov. ser.), 25(67):2 (1949), 307–314 | Zbl

[1950] “Nekotorye otsenki dlya ogranichennykh funktsii”, Mat. sb. (Nov. ser.), 26(68):1 (1950), 7–18 | Zbl

[1950a] “O tipichno veschestvennykh funktsiyakh”, Mat. sb. (Nov. ser.), 27(69):2 (1950), 201–218 | Zbl

[49] “K teorii odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 28(70):2 (1951), 351–358 | Zbl

[50] “K teorii odnolistnykh funktsii”, Mat. sb. (Nov. ser.), 29(71):1 (1951), 197–208 | Zbl

[51] “O mazhoratsii podchinennykh analiticheskikh funktsii. I”, Mat. sb. (Nov. ser.), 29(71):1 (1951), 209–224 | Zbl

[52] “O podchinennykh odnolistnykh funktsiyakh”, Tr. Mat. in-ta AN SSSR, 38, 1951, 68–71 | Zbl

[53] “Metod variatsii v konformnom otobrazhenii. IV”, Mat. sb. (Nov. ser.), 29(71):2 (1951), 455–468 | Zbl

[54] “O parametricheskom predstavlenii funktsii, odnolistnykh v koltse”, Mat. sb. (Nov. ser.), 29(71):2 (1951), 469–476 | Zbl

[55] “O mazhoratsii podchinennykh analiticheskikh funktsii. II”, Mat. sb. (Nov. ser.), 29(71):3 (1951), 593–602 | Zbl

[56] “O zadache koeffitsientov odnolistnykh funktsii”, Dokl. AN SSSR, 81:5 (1951), 721–723 | Zbl

[57] “Ob odnom metode variatsii v teorii analiticheskikh funktsii”, Uchen. zap. LGU, 1952, no. 144, 85–101

[58] Geometricheskaya teoriya funktsii kompleksnogo peremennogo, GITTL, M.–L., 1952, 540 pp.

[A] Aleksandrov I. A., Parametricheskie prodolzheniya v teorii odnolistnykh funktsii, «Nauka», M., 1976 | MR

[Ai] Aizenberg L. A., Formuly Karlemana v kompleksnom analize, «Nauka», Novosibirsk, 1990 | MR | Zbl

[AlKuzL] Alenitsyn Yu. E., Kuzmina G. V., Lebedev N. A., “Metody i rezultaty geometricheskoi teorii funktsii”, Dobavlenie k knige: Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, 2-e izd., «Nauka», M., 1966, 542–626 | MR

[B] Bakhtina G. P., Variatsionnye metody i kvadratichnye differentsialy v zadachakh o nenalegayuschikh oblastyakh, Avtoref. kand. diss., Kiev, 1975, 12 pp.

[BaSo] Barnard R. W., Solynin A. Yu., “Local variations and minimal area problem for Caratheodory functions”, Indiana Univ. Math. J., 53:1 (2004), 135–167 | DOI | MR | Zbl

[BaPSo1] Barnard R. W., Pearce K., Solynin A. Yu., “An isoperimetric inequality for logarithmic capacity”, Ann. Acad. Sci. Fenn. Math., 27:2 (2002), 419–436 | MR | Zbl

[BaPSo2] Barnard R. W., Pearce K., Solynin A. Yu., “Area, width, and logarithmic capacity of convex sets”, Pacific J. Math., 212:1 (2003), 13–23 | MR | Zbl

[BarKh] Bart V. A., Khavin V. P., “Teoremy Segë–Kolmogorova–Kreina o vesovoi approksimatsiii i formuly karlemanovskogo tipa”, Ukr. mat. Zhurnal, 46:1 (1994), 100–127 | MR | Zbl

[Br1] de Branges L., A proof of the Bieberbach conjecture, LOMI Preprints, no. E-5–84, Leningrad, 1984

[Br2] de Branges L., “A proof of the Bieberbach conjecture”, Acta Math., 154 (1985), 137–152 | DOI | MR | Zbl

[D1] Dubinin V. N., “Ob izmenenii garmonicheskoi mery pri simmetrizatsii”, Mat. sb., 124:2 (1984), 272–279 | MR | Zbl

[D3] Dubinin V. N., “Preobrazovanie funktsii i printsip Dirikhle”, Mat. zametki, 38:1 (1985), 49–55 | MR | Zbl

[D4] Dubinin V. N., “Razdelyayuschee preobrazovanie oblastei i zadachi ob ekstremalnom razbienii”, Zap. nauchn. semin. POMI, 168, 1988, 48–66 | MR | Zbl

[D5] Dubinin V. N., O maksimume odnogo konformnogo invarianta, Preprint DVO AN SSSR, In-t priklad. mat., Vladivostok, 1990 | MR

[D6] Dubinin V. N., “Simmetrizatsiya v teorii funktsii kompleksnogo peremennogo”, Uspekhi mat. nauk, 49:1 (1994), 3–76 | MR | Zbl

[D7] Dubinin V. N., “Konformnye otobrazheniya i neravenstva dlya algebraicheskikh polinomov”, Algebra i analiz, 13:5 (2001), 16–43 | MR | Zbl

[Du] Duren P., Univalent functions, Grundlehren Math. Wiss., 259, Springer-Verlag, New York–Berlin, 1983 | MR | Zbl

[E] Emelyanov E. G., “Nekotorye svoistva modulei semeistv krivykh”, Zap. nauchn. semin. POMI, 144, 1988, 72–82 | MR

[EKu] Emelyanov E. G., Kuzmina G. V., “Teoremy ob ekstremalnom razbienii v semeistvakh sistem oblastei razlichnykh vidov”, Zap. nauchn. semin. POMI, 237, 1997, 74–104 | MR | Zbl

[F1] Fedorov S. I., “O maksimume odnogo konformnogo invarianta”, Zap. nauchn. semin. LOMI, 112, 1981, 172–182 | MR

[F2] Fedorov S. I., “O variatsionnoi probleme Chebotareva v teorii emkosti ploskikh mnozhestv i teoremakh pokrytiya dlya odnolistnykh konformnykh otobrazhenii”, Mat. sb., 124:1 (1984), 121–139 | MR | Zbl

[FiPom] Fitzgerald C. H., Pommerenke Ch., “The de Branges theorem on univalent functions”, Trans. Amer. Math. Soc., 290:2 (1985), 683–690 | DOI | MR | Zbl

[FoKuz] Fomenko O. M., Kuz'mina G. V., “The last 100 days of the Bieberbach conjecture”, Math. Intelligencer, 8:1 (1986), 40–47 | DOI | MR | Zbl

[G] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, GITTL, M., 1952 ; German transl. VEB Deutscher Verlag Wiss., Berlin, 1957 ; 2-е изд., Наука, М., 1966; English trans. of 2th ed. Trans. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969 | MR | Zbl | MR

[Gr] Grunsky H., Lectures on theory of functions in multiply connected domains, Vandenhoeck and Reprecht, Gottingen, 1978 | MR | Zbl

[H] Hayman W. K., Multivalent functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1958 ; Kheiman V. K., Mnogolistnye funktsii, M., 1962 | MR | Zbl

[J1] Jenkins J. A., “On a problem of Gronwall”, Ann. of Math. (2), 59:3 (1954), 490–504 | DOI | MR | Zbl

[J2] Jenkins J. A., “Some theorems on boundary distortion”, Trans. Amer. Math. Soc, 81:2 (1956), 477–500 | DOI | MR | Zbl

[J4] Jenkins J. A., Univalent functions and conformal mappings, Ergeb. Math. Grenzgeb. (N. F.), 18, Springer-Verlag, Berlin etc., 1958 ; 2nd ed. corrected, 1965; Dzhenkins Dzh., Odnolistnye funktsii i konformnye otobrazheniya, M., 1962 | MR | Zbl

[J3] Jenkins J. A., “On the existence of certain extremal metric”, Ann. Math., 66:3 (1957), 440–453 ; Tohoku Math. J. (2), 45:2 (1993), 247–257 | DOI | MR | Zbl | DOI | MR

[J5] Jenkins J. A., “On certain geometrical problems associated with capacity”, Math. Nachr., 39:4–6 (1969), 349–356 | DOI | MR | Zbl

[J6] Jenkins J. A., “The method of extremal metric”, The Bieberbach Conjecture (West Lafayette Ind., 1985), Math. Surveys Monographs, 21, Amer. Math. Soc., Providence, RI, 1986, 95–104 | MR

[K1] Kolbina L. I., “Nekotorye ekstremalnye zadachi v konformnom otobrazhenii”, Dokl. AN SSSR, 84 (1952), 865–868 | MR | Zbl

[K2] Kolbina L. I., “Konformnoe otobrazhenie edinichnogo kruga na nenalegayuschie drug na druga oblasti”, Vestn. Leningr. un-ta, ser. fiz. mat. khim., 1955, no. 5, 37–43 | MR | Zbl

[KrKuh] Kvazikonformnye otobrazheniya — novye metody i prilozheniya, «Nauka», Novosibirsk, 1984

[Ku] Kuznetsov V. O., “O svoistvakh assotsiirovannykh kvadratichnykh differentsialov v nekotorykh ekstremalnykh zadachakh”, Zap. nauchn. semin. POMI, 168, 1988, 85–97 | MR | Zbl

[Kuz1] Kuzmina G. V., “Moduli semeistv krivykh i kvadratichnye differentsialy”, Tr. Mat. in-ta AN SSSR, 139, 1980, 3–241 | MR

[Kuz2] Kuzmina G. V., “K voprosu o maksimume proizvedeniya konformnykh radiusov nenalegayuschikh oblastei”, Zap. nauchn. semin. LOMI, 100, 1980, 131–145 | MR | Zbl

[Kuz3] Kuzmina G. V., “K voprosu ob ekstremalnom razbienii rimanovoi sfery”, Zap. nauchn. semin. POMI, 198, 1990, 72–95

[Kuz4] Kuzmina G. V., “Metody geometricheskoi teorii funktsii. I, II”, Algebra i analiz, 9:3 (1997), 41–103 | MR | Zbl

[Kuz5] Kuzmina G. V., “Metod ekstremalnoi metriki v zadachakh o maksimume proizvedeniya stepepenei konformnykh radiusov nenalegayuschikh oblastei pri nalichii svobodnykh parametrov”, Zap. nauchn. semin. POMI, 302, 2003, 52–67 | MR

[L1] Lebedev N. A., “Mazhorantnaya oblast dlya vyrazhenniya $I=log z^\lambda f'(z)^{1-\lambda}f(z)^{-\lambda}$ v klasse $S$”, Vestn. Leningr. un-ta, ser. mat. fiz. khim., 1955, no. 3, 29–41 | Zbl

[L2] Lebedev N. A., “Nekotorye otsenki dlya funktsii, regulyarnykh i odnolistnykh v kruge”, Vestn. Leningr. un-ta, ser. mat. fiz. khim., 1955, no. 4, 3–21 | Zbl

[L3] Lebedev N. A., Printsip ploschadei v teorii odnolistnykh funktsii, «Nauka», M., 1975 | MR | Zbl

[Le] Leung Y. J., “On the Nth diameter problem in the class $\Sigma$”, Complex Variables Theory Appl., 9:2–3 (1987 \), 227–239 | MR | Zbl

[M] Milin I. M., Odnolistnye funktsii i ortonormirovannye sistemy, «Nauka», M., 1971 | MR | Zbl

[P] Partington R. J., Interpolation, identification, and sampling, Clarendon Press, Oxford, 1997 | MR | Zbl

[Pi] Pirl U., “Über die geometrische Gestalt eines Extremalkontinuums aus der Theorie der konformer Abbildungen”, Math. Nachr., 39:1–6 (1969), 297–312 | DOI | MR | Zbl

[PoSz] Polya G., Szego G., Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, Princeton, NJ, 1951 ; Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatgiz, M., 1962 | MR | Zbl

[Pom1] Pommerenke Ch., “Über die Subordination analytisher Function”, J. Reine Angew. Math., 218 (1965), 159–173 | MR | Zbl

[Pom2] Pommerenke Ch., “On a variational method for univalent functions”, Michigan Math. J., 17:1 (1970), 1–3 | DOI | MR | Zbl

[Pom3] Pommerenke Ch., Univalent functions, with a chapter on quadratic differentials by G. Jensen, Math. Lehrbucher, 25, Vandenhoeck and Ruprecht, Gottingen, 1975 | MR

[RS] Reich E., Schiffer M., “Estimates for the transfinite diameter of a continuum”, Math. Z., 85:1 (1964), 91–106 | DOI | MR | Zbl

[SchaSp] Schaeffer A. C., Spenser D. C., “Coefficient regions for schlicht functions”, Amer. Math. Soc. Colloq. Publ., 35 (1950), Amer. Math. Soc., New York | MR

[SSp] Schiffer M., Spencer D. C., Functionals on finite Riemann surfaces, Princeton Univ. Press, Princeton, NJ, 1954 ; Shiffer M., Spenser D. K., Funktsionaly na konechnykh rimanovykh poverkhnostyakh, IL, M., 1957 | MR | Zbl

[So1] Solynin A. Yu., “Zavisimost problemy modulya dlya semeistva neskolkikh klassov krivykh ot parametrov”, Zap. nauchn. semin. POMI, 144, 1985, 136–146 | MR

[So2] Solynin A. Yu., “Reshenie odnoi izoperimetricheskoi zadachi Polia–Sege”, Zap. nauchn. semin. LOMI, 168, 1988, 140–153 | MR | Zbl

[So3] Solynin A. Yu., “Moduli i ekstremalno-metricheskie problemy”, Algebra i analiz, 11:1 (1999), 3–86 | MR | Zbl

[SoZ] Solynin A. Yu., Zalgaller V. A., “An isoperimetric inequality for logarithmic capacity of polygons”, Ann. of Math. (2), 159:1 (2004), 277–303 | DOI | MR | Zbl

[St] Strebel K., Quadratic differentials, Ergeb. Math. Grenzgeb. (3), 5, Springer-Verlag, Berlin–New York, 1984 | MR | Zbl