Abstract
We obtain numerous new estimates for the harmonic measures of continua concerning several problems, which have been considered previously by T. Hall, D. Gaier, J. A. Jenkins, W. K. Hayman and J.-M. Wu, and others. We usually give a complete solution of a problem, including a description of the extremal configurations. Our approach is based on some new results in the Jenkins Module theory and in the theory of symmetrization.
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References
A. Baernstein II,On the harmonic measure of slit domains, Complex Variables9 (1987), 131–142.
A. Baernstein II,An extremal property of meromorphic functions with n-fold symmetry, Complex Variables21 (1993), 137–148.
K. Barth, D. Brannan and W. Hayman,Research problems in complex analysis, Bull. London Math. Soc.16 (1984), 490–517.
A. Beurling,Études sur un problème de majoration, Thèse, Uppsala, 1933.
V. N. Dubinin,On the change in harmonic measure under symmetrization (in Russian), Mat. Sb.124 (1984), 272–279; English transl. in Math. USSR Sbornik52 (1985), 267–273.
V. N. Dubinin,Transformation of functions and the Dirichlet principle (in Russian), Mat. Zametki38 (1985), 49–55; English transl. in Math. Notes38 (1985), 539–542.
V. N. Dubinin,Capacities and geometric transformations of subsets in n-space, Geom. Funct. Anal.3 (1993), 342–369.
V. N. Dubinin,Symmetrization in geometric theory of functions of a complex variable (in Russian), Uspehi Mat. Nauk49 (1994), 3–76.
E. G. Emel’yanov,Some properties of the moduli of families of curves (in Russian), Zap. Nauchn. Semin. LOMI144 (1985), 72–82; English transl. in J. Soviet Math.38 (1987), 2081–2090.
E. G. Emel’yanov,Problems of extremal decomposition (in Russian), Zap. Nauchn. Semin. LOMI154, (1986), 76–89; English transl. in J. Soviet Math.43 (1988), 2558–2566.
C. H. FitzGerald, B. Rodin and S. E. Warschawski,Estimates of the harmonic measure of a continuum in the unit disk, Trans. Amer. Math. Soc.287 (1985), 681–685.
D. Gaier,Estimates of conformal mappings near the boundary, Indiana Univ. Math. J.21 (1972), 581–595.
D. Gaier,A note on Hall’s lemma, Proc. Amer. Math. Soc.37 (1973), 97–99.
M. N. Gavril’juk and A. Yu. Solynin,Applications of the Module Theory to Extremal Problems (in Russian), Kuban University, Krasnodar, 1983. Deponirovano in VINITI, no. 3072, 139p.
D. Hamilton,Covering theorems for univalent functions, Math. Z.191 (1986), 327–349.
W. K. Hayman and J.-M. G. Wu,Level sets of univalent functions, Comment. Math. Helv.56 (1981), 366–403.
J. Hersch,Longueurs extrémales et théorie des fonctions, Thèse (Orell Füssli Arts Graphiques s.A.), Zürich, 1955.
F. Huckemann,An extremal problem for harmonic measure, Comment. Math. Helv.52 (1977), 611–624.
S. Huckemann,Ein Extremalproblem für das harmonische mass einer Familie von Kontinua im Einheitskreis, Mitt. Math. Sem. Giessen184 (1988), 1–64.
J. A. Hummel and B. Pinchuk,Harmonic measure on an annulus, Complex Variables21 (1993), 243–252.
J. A. Jenkins,On the existence of certain general extremal metrics, Ann. of Math. (2)66 (1957), 440–453.
J. A. Jenkins,Univalent Functions and Conformal Mapping, Springer-Verlag, Berlin, 1958.
J. A. Jenkins,On the Denjoy conjecture, Canad. J. Math.10 (1958), 627–631.
J. A. Jenkins,On a problem concerning harmonic measure, Math. Z.135 (1974), 279–283.
J. A. Jenkins,On a lemma of Tord Hall, Bull. Inst. Math. Acad. Sinica2 (1974), 371–375.
J. A. Jenkins,The method of the extremal metric, inThe Biberbach Conjecture (West Lafayette, Ind., 1985), Math. Surveys Monographs,21, Amer. Math. Soc., Providence, R.I., 1986, pp. 95–104.
J. A. Jenkins,Some estimates for harmonic measure, inComplex Analysis 1, Proceedings of the special year held at the University of Maryland (College Park, 1985–86), Lecture Notes in Math.1275, Springer-Verlag, New York, 1987, pp. 210–214.
J. A. Jenkins,Some estimates for harmonic measures. II, Proc. Amer. Math. Soc.111 (1991), 441–442.
J. A. Jenkins,Some estimates for harmonic measures. III, Proc. Amer. Math. Soc.119 (1993), 199–201.
G. V. Kuz'mina,Moduli of families of curves and quadratic differentials (in Russian), Trudy Matematicheskogo Instituta imeni V. A. Steklova139 (1980); English transl. in Proc. Steklov Inst. Math.1 (1982).
G. V. Kuz'mina,Extremal properties of quadratic differentials with strip-shaped domains in the structure of the trajectories (in Russian), Zap. Nauchn. Semin. LOMI154 (1986), 110–129; English transl. in J. Soviet Math.43 (1988), 2579–2591.
L. Liao,Certain extremal problems concerning module and harmonic measure, J. Analyse Math.40 (1981), 1–42.
D. E. Marshall and C. Sundberg,Harmonic measure and radial projection, Trans. Amer. Math. Soc.316 (1989), 81–95.
U. Pirl,Über die geometrische Gestalt eines Extremalkontinuums aus der Theorie der konformen Abbildung, Math. Nachr.39 (1969), 297–312.
A. Yu. Solynin,Extremal decompositions of the plane or disk into two nonoverlapping domains (in Russian), Kuban University, Krasnodar, 1984. Deponirovano in VINITI, no. 7800, 16p.
A. Yu. Solynin,The dependence on parameters of the modulus problem for families of several classes of curves (in Russian), Zap. Nauchn. Semin. LOMI144 (1985), 136–145; English transl. in J. Soviet Math.38 (1987), 2131–2139.
A. Yu. Solynin,Harmonic measure of continua with a fixed diameter (in Russian), Zap. Nauchn. Semin. LOMI144 1985, 146–149; English transl. in J. Soviet Math.38 (1987), 2140–2142.
A. Yu. Solynin,Some extremal problems for pairs of functions with no common values (in Russian), Zap. Nauchn. Semin. LOMI160 (1987), 159–169; English transl. in J. Soviet Math.52 (1990), 3115–3124.
A. Yu. Solynin,Solution of a Polya-Szegö isoperimetric problem (in Russian), Zap. Nauchn. Semin. LOMI168 (1988), 140–153; English transl. in J. Soviet Math.53 (1991), 311–320.
A. Yu. Solynin,Geometric properties of extremal decompositions and estimates for the modulus of families of curves in an annulus (in Russian), Zap. Nauchn. Semin. POMI204 (1993), 93–114.
A. Yu. Solynin,The boundary distortion and extremal problems in certain classes of univalent functions (in Russian), Zap. Nauchn. Semin. POMI204 (1993), 115–142.
A. Yu. Solynin,Decompositions into nonoverlapping domains and extremal properties of univalent functions (in Russian), Zap. Nauchn. Semin. POMI212 (1994), 139–163.
A. Yu. Solynin,Extremal configurations for some problems on the capacity and harmonic measure (in Russian), Zap. Nauchn. Semin. POMI226 (1996), 170–195.
A. Yu. Solynin and M. Vuorinen,Extremal problems and symmetrization for ring domains in the plane, Trans. Amer. Math. Soc.348 (1996), 4095–4112.
A. Yu. Solynin,Moduli and Extremal-Metric problems (in Russian), Preprint POMI3/1997, St. Petersburg, 1997, 67 pp.
A. E. Fryntov,On an estimate of harmonic measures, Dopovidi Akad. Nauk Ukraini10 (1994), 20–22.
D. E. Marshall and C. Sundberg,Harmonic measure of curves in the disk, J. Analyse Math.70 (1996), 175–224.
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Solynin, A.Y. Extremal problems on conformal moduli and estimates for harmonic measures. J. Anal. Math. 74, 1–49 (1998). https://doi.org/10.1007/BF02819444
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DOI: https://doi.org/10.1007/BF02819444