Geodesic
A survey on the integral means spectrum for conformal mappings
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 104-115 Cet article a éte moissonné depuis la source Math-Net.Ru

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The present survey is devoted to the description of the results about estimates of the integral means in various classes of analytic functions. Great importance is given to the boundary properties of conformal mappings. The systematic results on the estimation of the integral means spectrum and the well-known law of the iterated logarithm for conformal mappings, which was proven by N. G. Makarov in 1985, were the basic mathematical tools for this study. Precise estimates of the integral means in various subclasses of univalent functions are described. Estimates of the integral means spectrum in the class of functions mapping the exterior of the unit disk onto the attraction basins of the infinity of algebraic polynomials are presented.
Keywords: conformal mappings, integral means spectrum, harmonic measure. 
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I. R. Kayumov. A survey on the integral means spectrum for conformal mappings. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 2, pp. 104-115. http://geodesic.mathdoc.fr/item/UZKU_2015_157_2_a8/

[1] Littlewood J., “On inequalities in the theory of functions”, Proc. London Math. Soc. (2), 23:1 (1925), 481–519 | DOI | MR | Zbl

[2] Gronwall T. H., “Some remarks on conformal representation”, Ann. of Math. -, 16:1/4 (1914–1915), 72–76 | DOI | MR | Zbl

[3] Clunie J., Pommerenke Ch., “On the coefficients of univalent functions”, Michigan Math. J., 14:1 (1967), 71–78 | DOI | MR | Zbl

[4] Carleson L., Jones P. W., “On coefficient problems for univalent functions and conformal dimension”, Duke Math. J., 66:2 (1992), 169–206 | DOI | MR | Zbl

[5] Makarov N. G., “Fine structure of harmonic measure”, St. Petersburg Math. J., 10:2 (1999), 217–268 | MR | Zbl

[6] Pommerenke Ch., “On the integral means of the derivative of a univalent function”, J. London Math. Soc. (2), 32:2 (1985), 254–258 | DOI | MR | Zbl

[7] Hedenmalm H., Shimorin S., “On the universal integral means spectrum of conformal mappings near the origin”, Proc. Amer. Math. Soc., 135:7 (2007), 2249–2255 | DOI | MR | Zbl

[8] Makarov N. G., “A note on the integral means of the derivative in conformal mapping”, Proc. Amer. Math. Soc., 96:2 (1986), 233–236 | DOI | MR | Zbl

[9] Pommerenke Ch., Boundary Behaviour of Conformal Maps, Springer, Berlin–Heidelberg, 1992, 300 pp. | MR | Zbl

[10] Kayumov I. R., “Lower estimates for integral means of univalent functions”, Ark. Mat., 44:1 (2006), 104–110 | DOI | MR | Zbl

[11] Kraetzer Ph., “Experimental bounds for the universal integral means spectrum of conformal maps”, Complex Var. Theory Appl., 31:4 (1996), 305–309 | DOI | MR | Zbl

[12] Beliaev D., Smirnov S., “Random conformal snowflakes”, Ann. of Math. (2), 172:1 (2010), 597–615 | DOI | MR | Zbl

[13] Kayumov I. R., Maklakov D. V., Kayumov F. D., “Otsenki spektra integralnykh srednikh dlya lakunarnykh ryadov”, Izv. vuzov. Matem., 2014, no. 10, 79–85 | Zbl

[14] Carleson L., Makarov N. G., “Some results connected with Brennan's conjecture”, Ark. Mat., 32:1 (1994), 33–62 | DOI | MR | Zbl

[15] Feng J., MacGregor T. H., “Estimates of integral means of the derivatives of univalent functions”, J. Anal. Math., 29:1 (1976), 203–231 | DOI | Zbl

[16] Kayumov I. R., “Ob odnom neravenstve dlya universalnogo spektra integralnykh srednikh”, Matem. zametki, 84:1 (2008), 139–143 | DOI | MR | Zbl

[17] Makarov N. G., “Conformal mapping and Hausdorff measures”, Ark. Mat., 25:1–2 (1987), 41–89 | DOI | MR | Zbl

[18] Makarov N. G., “On the distortion of boundary sets under conformal mappings”, Proc. London Math. Soc. (3), 51:3 (1985), 369–384 | DOI | MR | Zbl

[19] Kayumov I. R., “Spektr integralnykh srednikh i zakon povtornogo logarifma dlya konformnykh otobrazhenii”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 148, no. 2, 2006, 85–96 | Zbl

[20] Kayumov I. R., “The law of the iterated logarithm for locally univalent functions”, Ann. Acad. Sci. Fennicae, 27:2 (2002), 357–364 | MR | Zbl

[21] Hedenmalm H., Kayumov I. R., “On the Makarov law of the iterated logarithm”, Proc. Amer. Math. Soc., 135:7 (2007), 2235–2248 | DOI | MR | Zbl

[22] Kayumov I. R., “K zakonu povtornogo logarifma dlya konformnykh otobrazhenii”, Matem. zametki, 79:1 (2006), 150–153 | DOI | MR | Zbl

[23] Lavrentev M. A., “O nekotorykh granichnykh zadachakh v teorii odnolistnykh funktsii”, Matem. sb., 1(43):6 (1936), 815–846 | Zbl

[24] Kayumov I. R., “Metricheskie svoistva garmonicheskoi mery na zhordanovykh krivykh”, Mezhdunar. konf. “Funktsionalnye prostranstva i teoriya approksimatsii”, posvyasch. 110-letiyu so dnya rozhd. S. M. Nikolskogo, Tez. dokl., M., 2015, 156

[25] McMullen C. T., “Thermodynamics, dimension and the Weil–Petersson metric”, Invent. Math., 173:2 (2008), 365–425 | DOI | MR | Zbl

[26] Le Thanh Hoang N., Zinsmeister M., “On Minkowski dimension of Jordan curves”, Ann. Acad. Sci. Fenn. Math., 39:2 (2014), 787–810 | MR | Zbl

[27] Kayumov I. R., “The integral means spectrum for lacunary series”, Ann. Acad. Sci. Fennicae, 26:2 (2001), 447–453 | MR | Zbl

[28] Milnor Dzh., Golomorfnaya dinamika, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2000, 320 pp.

[29] Kayumov I. R., “Granichnoe povedenie analiticheskikh proizvedenii Rissa v kruge”, Matem. trudy, 9:1 (2006), 34–51 | MR | Zbl

[30] Littlewood J., “On the coefficients of schlicht functions”, Quart. J. Math. Oxford Ser., 9:1 (1938), 14–20 | DOI | MR | Zbl

[31] Kayumov I. R., “Teorema iskazheniya dlya odnolistnykh lakunarnykh ryadov”, Sib. matem. zhurn., 44:6 (2003), 1273–1279 | MR | Zbl

[32] Kraetzer Ph., Algorithmische Methoden der konformen Abbildungen auf fraktale Gebiete, Dissertation, Berlin, 2000, 60 pp.

[33] Kayumov I. R., “O gipoteze Brennana dlya spetsialnogo klassa funktsii”, Matem. zametki, 78:4 (2005), 537–541 | DOI | MR | Zbl

[34] Brennan J. E., “The integrability of the derivative in conformal mapping”, J. London Math. Soc. (2), 18:2 (1978), 261–272 | DOI | MR | Zbl

[35] Kayumov F. D., “Integralnye otsenki dlya proizvodnykh odnolistnykh funktsii”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 155, no. 2, 2013, 83–90

[36] Kayumov I. R., “Tochnye otsenki integralnykh srednikh dlya trekh klassov oblastei”, Matem. zametki, 76:4 (2004), 510–516 | DOI | MR | Zbl