Abstract
In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structures, making it hard to describe them and to attack such problems. This is particularly true for questions related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: random conformal snowflakes, and investigate their properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behavior of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.

[Beliaev_comp] D. Beliaev, "Integral means spectrum of random conformal snowflakes," Nonlinearity, vol. 21, iss. 7, pp. 14351442, 2008.
@article {Beliaev_comp, MRKEY = {2425327},
AUTHOR = {Beliaev, D.},
TITLE = {Integral means spectrum of random conformal snowflakes},
JOURNAL = {Nonlinearity},
FJOURNAL = {Nonlinearity},
VOLUME = {21},
YEAR = {2008},
NUMBER = {7},
PAGES = {14351442},
ISSN = {09517715},
CODEN = {NONLE5},
MRCLASS = {37F10 (28A80 30C50 37C45)},
MRNUMBER = {2009g:37041},
DOI = {10.1088/09517715/21/7/003},
ZBLNUMBER = {1154.30018},
} 
[Bthesis] D. Beliaev, "Harmonic measure on random fractals," PhD Thesis , Royal Institute of Technology, 2005.
@phdthesis{Bthesis,
author = {Beliaev, D.},
TITLE = {Harmonic measure on random fractals},
YEAR={2005},
SCHOOL={Royal Institute of Technology},
} 
[BeSmECM] D. Beliaev and S. Smirnov, "Harmonic measure on fractal sets," in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 4159.
@incollection {BeSmECM, MRKEY = {2185735},
AUTHOR = {Beliaev, D. and Smirnov, S.},
TITLE = {Harmonic measure on fractal sets},
BOOKTITLE = {European {C}ongress of {M}athematics},
PAGES = {4159},
PUBLISHER = {Eur. Math. Soc., Zürich},
YEAR = {2005},
MRCLASS = {31C20 (28A80 30C50 37F35)},
MRNUMBER = {2007d:31013},
MRREVIEWER = {Klaus G{ü}rlebeck},
ZBLNUMBER = {1079.30026},
} 
[CaJo] L. Carleson and P. W. Jones, "On coefficient problems for univalent functions and conformal dimension," Duke Math. J., vol. 66, iss. 2, pp. 169206, 1992.
@article {CaJo, MRKEY = {1162188},
AUTHOR = {Carleson, Lennart and Jones, Peter W.},
TITLE = {On coefficient problems for univalent functions and conformal dimension},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {66},
YEAR = {1992},
NUMBER = {2},
PAGES = {169206},
ISSN = {00127094},
CODEN = {DUMJAO},
MRCLASS = {30C50 (30C85 30D05)},
MRNUMBER = {93c:30022},
MRREVIEWER = {Ch. Pommerenke},
DOI = {10.1215/S0012709492066051},
ZBLNUMBER = {0765.30005},
} 
[deBranges] L. de Branges, "A proof of the Bieberbach conjecture," Acta Math., vol. 154, iss. 12, pp. 137152, 1985.
@article {deBranges, MRKEY = {772434},
AUTHOR = {de Branges, Louis},
TITLE = {A proof of the {B}ieberbach conjecture},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {154},
YEAR = {1985},
NUMBER = {12},
PAGES = {137152},
ISSN = {00015962},
CODEN = {ACMAA8},
MRCLASS = {30C50},
MRNUMBER = {86h:30026},
MRREVIEWER = {P. L. Duren},
DOI = {10.1007/BF02392821},
ZBLNUMBER = {0573.30014},
} 
[HJKPS] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, "Fractal measures and their singularities: the characterization of strange sets," Phys. Rev. A, vol. 33, iss. 2, pp. 11411151, 1986.
@article {HJKPS, MRKEY = {823474},
AUTHOR = {Halsey, Thomas C. and Jensen, Mogens H. and Kadanoff, Leo P. and Procaccia, Itamar and Shraiman, Boris I.},
TITLE = {Fractal measures and their singularities: the characterization of strange sets},
JOURNAL = {Phys. Rev. A},
FJOURNAL = {Physical Review. A. Third Series},
VOLUME = {33},
YEAR = {1986},
NUMBER = {2},
PAGES = {11411151},
ISSN = {10502947},
CODEN = {PLRAAN},
MRCLASS = {58F13 (82A05)},
MRNUMBER = {87h:58125a},
MRREVIEWER = {Thomas D. Rogers},
DOI = {10.1103/PhysRevA.33.1141},
} 
[HeSh] H. Hedenmalm and S. Shimorin, "Weighted Bergman spaces and the integral means spectrum of conformal mappings," Duke Math. J., vol. 127, iss. 2, pp. 341393, 2005.
@article {HeSh, MRKEY = {2130416},
AUTHOR = {Hedenmalm, H{\aa}kan and Shimorin, Serguei},
TITLE = {Weighted {B}ergman spaces and the integral means spectrum of conformal mappings},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {127},
YEAR = {2005},
NUMBER = {2},
PAGES = {341393},
ISSN = {00127094},
CODEN = {DUMJAO},
MRCLASS = {30C40 (30C85 46E15)},
MRNUMBER = {2005m:30010},
MRREVIEWER = {Ke He Zhu},
DOI = {10.1215/S0012709404127253},
ZBLNUMBER = {1075.30005},
} 
[Littlewood25] J. E. Littlewood, "On inequalities in the theory of functions," Proc. L.M.S., vol. 23, pp. 481519, 1925.
@article{Littlewood25,
author={Littlewood, J. E.},
TITLE={On inequalities in the theory of functions},
JOURNAL={Proc. L.M.S.},
VOLUME={23},
PAGES={481519},
YEAR={1925},
JFMNUMBER={51.0247.03}
} 
[Makarov] N. G. Makarov, "Fine structure of harmonic measure," Algebra i Analiz, vol. 10, iss. 2, pp. 162, 1998.
@article {Makarov, MRKEY = {1629379},
AUTHOR = {Makarov, N. G.},
TITLE = {Fine structure of harmonic measure},
JOURNAL = {Algebra i Analiz},
FJOURNAL = {Rossiĭskaya Akademiya Nauk. Algebra i Analiz},
VOLUME = {10},
YEAR = {1998},
NUMBER = {2},
PAGES = {162},
ISSN = {02340852},
MRCLASS = {30C85 (28D99 31A15)},
MRNUMBER = {2000g:30018},
MRREVIEWER = {Albert Baernstein, II},
ZBLNUMBER = {0909.30016},
} 
[MaPo] N. G. Makarov and C. Pommerenke, "On coefficients, boundary size and Hölder domains," Ann. Acad. Sci. Fenn. Math., vol. 22, iss. 2, pp. 305312, 1997.
@article {MaPo, MRKEY = {1469793},
AUTHOR = {Makarov, Nikolai G. and Pommerenke, Christian},
TITLE = {On coefficients, boundary size and {H}ölder domains},
JOURNAL = {Ann. Acad. Sci. Fenn. Math.},
FJOURNAL = {Academiæ Scientiarum Fennicæ . Annales. Mathematica},
VOLUME = {22},
YEAR = {1997},
NUMBER = {2},
PAGES = {305312},
ISSN = {00661953},
MRCLASS = {30C50},
MRNUMBER = {98i:30021},
MRREVIEWER = {K. S. Padmanabhan},
ZBLNUMBER={0890.30010},
} 
[Mandelbrot72] B. B. Mandelbrot, "Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence," in Statistical Models Turbulence, Proc. Sympos. Univ. California, San Diego, 1972, pp. 333351.
@inproceedings{Mandelbrot72,
author={Mandelbrot, B. B.},
TITLE={Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence},
BOOKTITLE={Statistical Models Turbulence, Proc. Sympos. Univ. California, San Diego},
VENUE={La Jolla, 1971},
SERIES={Lecture Notes Phys.},
VOLUME={12},
PAGES={333351},
YEAR={1972},
ZBLNUMBER={0227.76081},
} 
[Mandelbrot74] B. B. Mandelbrot, "Intermittent turbulence in selfsimilar cascades: divergence of high moments and dimension of the carrier," J. Fluid Mech., vol. 62, pp. 331358, 1974.
@article{Mandelbrot74,
author={Mandelbrot, B. B.},
TITLE={Intermittent turbulence in selfsimilar cascades: divergence of high moments and dimension of the carrier},
JOURNAL={J. Fluid Mech.},
VOLUME={62},
YEAR={1974},
PAGES={331358},
} 
[Pommerenke67lms] C. Pommerenke, "On the coefficients of univalent functions," J. London Math. Soc., vol. 42, pp. 471474, 1967.
@article {Pommerenke67lms, MRKEY = {0222277},
AUTHOR = {Pommerenke, Christian},
TITLE = {On the coefficients of univalent functions},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {42},
YEAR = {1967},
PAGES = {471474},
ISSN = {00246107},
MRCLASS = {30.43},
MRNUMBER = {36 \#5329},
MRREVIEWER = {F. R. Keogh},
DOI = {10.1112/jlms/s142.1.471},
ZBLNUMBER = {0177.33602},
} 
[Pommerenke75] C. Pommerenke, Univalent Functions, Göttingen: Vandenhoeck & Ruprecht, 1975.
@book {Pommerenke75, MRKEY = {0507768},
AUTHOR = {Pommerenke, Christian},
TITLE = {Univalent Functions},
PUBLISHER = {Vandenhoeck \& Ruprecht},
ADDRESS = {Göttingen},
YEAR = {1975},
PAGES = {376},
MRCLASS = {30A36},
MRNUMBER = {58 \#22526},
MRREVIEWER = {D. M. Campbell},
ZBLNUMBER = {0298.30014},
}