Skip to main content

Geometric function theory

  • Chapter
  • First Online:
Linear and Complex Analysis Problem Book 3

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1574))

  • 961 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bishop C. J., Brownian motion in Denjoy domains, Ann. Prob. (to appear).

    Google Scholar 

  2. Bishop C. J., A characterization of Poissonian domains, Arkiv Mat. (to appear).

    Google Scholar 

  3. Bishop C. J., Jones P. W., Harmonic measure and arclength, Ann. Math. 132 (1990), 511–547.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bishop C. J., Jones P. W., Harmonic measure, L2 estimates and the Swarzian derivative, preprint, 1990.

    Google Scholar 

  5. Bourgain J., On the Hausdorff dimension of harmonic measure in higher dimensions, Inv. Math 87 (1987), 477–483.

    Article  MathSciNet  MATH  Google Scholar 

  6. Jones P. W. and Wolff T. H., Hausdorff dimension of harmonic measures in the plane, Acta. Math. 161 (1988), 131–144.

    Article  MathSciNet  MATH  Google Scholar 

  7. Makarov N. G., On distortion of boundary set under conformal mappings, Proc. London Math. Soc. 51 (1985), 369–384.

    Article  MathSciNet  MATH  Google Scholar 

  8. Pommerenke Ch., On conformal mapping and linear measure, J. Analyse Math. 46 (1986), 231–238.

    Article  MathSciNet  MATH  Google Scholar 

  9. Wolff T. H., Counterexamples with harmonic gradients, preprint, 1987.

    Google Scholar 

References

  1. Korenblum B., AMS Abstracts, 855-30-04 (1990).

    Google Scholar 

  2. Korenblum B., A maximum principle for the Bergman space, Publications Matemátiques (Barcelona) (to appear).

    Google Scholar 

  3. Korenblum B., Transformations of zero sets by contractive operators in the Bergman space, Bull. Sc. Math. 2e série 114 (1990), 385–394.

    MathSciNet  MATH  Google Scholar 

  4. Korenblum B., Richards K., Majorization and domination in the Bergman space, preprint.

    Google Scholar 

  5. Korenblum B., O’Neil R., Richards K., Zhu K., Totally monotone functions with applications to the Bergman space, preprint.

    Google Scholar 

References

  1. Luecking D., Zhu K., Composition operators belonging to Schatten classes, Amer. J. Math (to appear).

    Google Scholar 

  2. MacCluer B., Shapiro J., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canadian J. Math. 38 (1986), 878–906.

    Article  MathSciNet  MATH  Google Scholar 

  3. Sarason D., Angular derivatives via Hilbert space, Complex Variables 10 (1988), 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  4. Shapiro J., The essential norm of a composition operator, Ann. Math. 12 (1987), 375–404.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Goodman A. W., Univalent functions, Vols. I and II, Polygonal Publishing House, Washington, New Jersey U.S.A., 1983.

    MATH  Google Scholar 

  2. Goodman A. W., Topics in mathematical analysis, World Scientific Publishing Co., Singapore, 1989.

    Google Scholar 

  3. Goodman A. W., Convex functions of bounded type, Proc. Amer. Math Soc. 92 (1984), 541–546.

    Article  MathSciNet  MATH  Google Scholar 

  4. Goodman A. W., More on convex functions of bounded type, Proc. Amer. Math Soc. 97 (1986). 303–306.

    Article  MathSciNet  MATH  Google Scholar 

  5. Goodman A. W., On uniformly starlike functions, Jour. of Math. Analysis and Appl. 155 (1991), 364–370.

    Article  MathSciNet  MATH  Google Scholar 

  6. Goodman A. W., On uniformly convex functions, Ann. Polo. Math. (to appear).

    Google Scholar 

  7. Ma Wancang, Mejia Diego, Minda David, Distortion theorems for euclidean k-convex functions, Complex Var. Theor. and Appl. (to appear).

    Google Scholar 

  8. Ma Wancang, Mejia Diego, Minda David, Distortion theorems for hyperbolically and spherically k-convex functions, Proc. Inter. Conf. New Trends in Geom. Func. Th. and Appl. (to appear).

    Google Scholar 

  9. Mejia Diego, Minda David, Hyperbolic geometry in k-convex regions, Pac. Jour. of Math. 141 (1990), 333–354.

    Article  MathSciNet  MATH  Google Scholar 

  10. Mejia Diego, Minda David, Hyperbolic geometry in spherically k-convex regions, Comp. Methods and Func. Th. Proc., Lecture notes in Math, vol. 1435, Springer, 1990.

    Google Scholar 

  11. Mejia Diego, Minda David, Hyperbolic geometry in hyperbolically k-convex regions, submitted.

    Google Scholar 

  12. Wirths K. J., Coefficient bounds for convex functions of bounded type, Proc. Amer. Math. Soc. 103 (1988), 525–530.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Krzyż J. G., Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 19–24.

    Article  MathSciNet  MATH  Google Scholar 

  2. Krzyż J. G., Harmonic analysis and boundary correspondence under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 225–242.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. de Branges L., A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bshouty D., Hengartner W., Criteria for the extremality of the Koebe mapping, Proc. Amer. Math. Soc. 111 (1991), no. 2, 403–411.

    Article  MathSciNet  MATH  Google Scholar 

  3. Duren P. L., Univalent Functions, Springer Verlag, Heidelberg-New York, 1983.

    MATH  Google Scholar 

  4. Golusin G. M., Geometric Theory of Functions of a Complex Variable, “Nauka”, Moscow, 1966 (Russian); English transl. Amer. Math. Soc., 1969.

    Google Scholar 

  5. Grinshpan A. Z., On the Taylor coefficients of certain classes of univalent functions, Metric Questions in the Theory of Functions, “Naukova Dumka”, Kiev, 1980, pp. 28–32. (Russian).

    Google Scholar 

  6. Grinshpan A. Z., On coefficients of powers of univalent functions, Sibirsk. Mat. Zh. 22 (1981), no. 4, 88–93 (Russian); English transl. in Siberian Math. J. 22 (1981), 551–555.

    MathSciNet  MATH  Google Scholar 

  7. Grinshpan A. Z., On the power stability for the Bieberbach inequality, Zap. Nauchn. Sem. LOMI 125 (1983), 58–64. (Russian).

    MathSciNet  MATH  Google Scholar 

  8. Grinshpan A. Z., Method of exponentiation for univalent functions, Theory of Functions and Applications, Proc. Conf. Saratov, 1988, Part 2, Izdat. Sar. Univ., Saratov, 1990, pp. 72–74. (Russian)

    Google Scholar 

  9. Grinshpan A. Z., Univalent functions with prescribed logarithmic restrictions, Annals Polonici Mathematici (to appear).

    Google Scholar 

  10. Milin I. M., Grinshpan A. Z., Logarithmic coefficients means of univalent functions, Complex Variables: Theory and Appl. 7 (1986), no. 1-3, 139–147.

    Article  MathSciNet  MATH  Google Scholar 

  11. Milin I. M., Univalent Functions and Orthonormal Systems, “Nauka”, Moscow, 1971 (Russian); English transl. Amer. Math. Soc., Providence, R.I., 1977.

    MATH  Google Scholar 

References

  1. De Branges L., Unitary linear systems whose transfer functions are Riemann mapping functions, Integral Equations and Operator Theory 19 (1986), 105–124.

    MathSciNet  MATH  Google Scholar 

  2. De Branges L., Underlying concepts in the proof of the Bieberbach conjecture, Proceedings of the International Congress of Mathematicians 1986, Berkeley, California, 1986, pp. 25–42.

    Google Scholar 

  3. De Branges L., Square Summable Power Series, Springer-Verlag, in preparation.

    Google Scholar 

  4. Li Kin Y., Rovnyak James, On the coefficients of Riemann mappings of the unit disk into itself, 1991, preprint.

    Google Scholar 

  5. Rovnyak J., Coefficient estimates for Riemann mapping functions, J. Anal. Math. 52 (1989), 53–93.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Tamrazov P. M., Smoothness and Polynomial Approximations, Naukova Dumka, Kiev, 1975. (Russian)

    MATH  Google Scholar 

  2. Näkki R., Palka B., Extremal length and Hölder continuity of conformal mappings, Comment. Math. Helvetici 61 (1986), 389–414.

    Article  MATH  Google Scholar 

  3. Belyî V. I., On moduli of continuity of exterior and interior conformal mappings of the unit disk, Ukranian Math. J. 41, (1989), no. 4, 469–475. (Russian).

    MathSciNet  Google Scholar 

References

  1. Baerstein A., An extremal problem for certain subharmonic functions in the plane, Rev. Mat. Iberoamericana 4 (1988), 199–219.

    Article  MathSciNet  Google Scholar 

  2. Cassels J. W. S., An introduction to the Geometry of numbers (2nd ed.), Springer, Berlin, 1972.

    MATH  Google Scholar 

  3. Fryntov A. E., An extremal problem of potential theory, Dokl. Akad. Nauk USSR 300 (1988), no. 4 (Russian); English transl. in Soviet Math.—Doklady 37 (1988), 754–755.

    Google Scholar 

  4. Hayman W. K., Subharmonic functions, vol. 2, Academic Press, London, 1989.

    MATH  Google Scholar 

  5. Hille E., Analytic Function Theory, vol. 2, Ginn, Boston, 1962.

    MATH  Google Scholar 

  6. Minda C. D., Bloch constants, J. Analyse Math. 41 (1982), 54–84.

    Article  MathSciNet  MATH  Google Scholar 

  7. Montgomery H., Minimal theta functions, Glasdow Math. J. 30 (1988), 75–83.

    Article  MathSciNet  MATH  Google Scholar 

  8. Osgood B., Phillips R., Sarnak P., Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.

    Article  MathSciNet  MATH  Google Scholar 

  9. Quine J. R., Heydari S. H., Song R. Y., Zeta regularized products, Trans. Amer. Math. Soc. (to appear).

    Google Scholar 

  10. Rogers C. A., Packing and Covering, Cambridge U. P., Cambridge, 1964.

    MATH  Google Scholar 

  11. Weitsman A., Symmetrization and the Poincaré metric, Annals of Math. 124 (1986), 159–169.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Abu-Muhanna Y., Lyzzaik A., The boundary behaviour of harmonic univalent maps, Pacific J. Math. 141 (1990), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  2. Clunie J., Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I 9 (1984), 3–25.

    MathSciNet  MATH  Google Scholar 

  3. Hengartner W., Schober G., Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473–483.

    Article  MathSciNet  MATH  Google Scholar 

  4. Hengartner W., Schober G., Curvature estimates for some minimal surfaces, Complex Analysis: Articles dedicated to Albert Pfluger on the occasion of his 80th birthday (Hersch J., Huber A., eds.), Birkhäuser Verlag, Basel, 1988, pp. 87–100.

    Google Scholar 

  5. Lewy H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692.

    Article  MathSciNet  MATH  Google Scholar 

  6. Nitsche J. C. C., On the module of double-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), 781–782.

    Article  MathSciNet  MATH  Google Scholar 

  7. Schober G., Planar harmonic mappings, Computational Methods and Function Theory, Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg, 1990, pp. 171–176.

    MATH  Google Scholar 

  8. Sheil-Small, T., Constants for planar harmonic mappings, J. London Math. Soc. 42 (1990), 237–248.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Alan F. Beardon and Kenneth Stephenson, The Schwarz-Pick lemma for circle packings, Ill. J. Math. 141 (1991), 577–606.

    MathSciNet  MATH  Google Scholar 

  2. Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J., 39 (1990), 1383–1425.

    Article  MathSciNet  MATH  Google Scholar 

  3. Koebe P., Kontaktprobleme der Konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164.

    MATH  Google Scholar 

  4. Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geometry 26 (1987), 349–360.

    MathSciNet  MATH  Google Scholar 

  5. Kenneth Stephenson, Circle packings in the approximation of conformal mappings, Bulletin, Amer. Math. Soc. (Research Announcements) 23 (1990), no. 2, 407–415.

    Article  MathSciNet  MATH  Google Scholar 

  6. Kenneth Stephenson, Thurston’s conjecture on circle packings in the nonhexagonal case, preprint.

    Google Scholar 

  7. William Thurston, The Geometry and Topology of 3-Manifolds, preprint, Princeton University Notes.

    Google Scholar 

References

  1. Milin I. M., On a property of logarithmic coefficients on univalent functions, Metric questions of the function theory, Naukova Dumka, Kiev, 1980, pp. 86–90. (Russian)

    MATH  Google Scholar 

  2. Lebedev N. A., An application of the area principle to problems on non overlapping domains, Trudy Mat. Inst. Akad. Nauk SSSR 60 (1961), 211–231. (Russian)

    Google Scholar 

  3. Milin I. M., On a conjecture on a logarithmic coefficients of univalent functions, Zapiski nauchn. semin. LOMI 125 (1983), 135–143 (Russian); English transl. in J. Soviet Math. 26 (1984), no. 6.

    MathSciNet  MATH  Google Scholar 

References

  1. Astala K., Fernández J. L., Rohde S. (1991) (to appear).

    Google Scholar 

  2. Astala K., Zinsmeister M., Teichmüller spaces and BMOA, Mittag-Leffler Report 20 (1989–90).

    Google Scholar 

  3. Bishop C. J., Jones P. W. Harmonic measure, L2-estimates and the Schwarzian derivative, preprint (1990).

    Google Scholar 

  4. Fernández J. L., Heinonen J., Martio O., Quasilines and conformal mappings, J. Analyse Math. 52 (1989), 117–132.

    Article  MathSciNet  MATH  Google Scholar 

  5. Garnett J. B., Gehring F. W., Jones P. W., Conformally invariant length sums, Indiana Univ. Math. J. 32 (1983), 809–829.

    Article  MathSciNet  MATH  Google Scholar 

  6. Haiman W. K., Wu J.-M., Level sets of univalent functions, Comment. Math. Helv. 56 (1981), 366–403.

    Article  MathSciNet  MATH  Google Scholar 

  7. Jones P. W., Marshall D. E., Critical points of Green’s function, harmonic measure and the corona problem, Ark. Math. 23 (1985), 281–314.

    Article  MathSciNet  MATH  Google Scholar 

  8. Øyma K., Harmonic measure and conformal length, Proc. Amer. Math. Soc. (to appear).

    Google Scholar 

  9. Väisälä J, Bounded turning and quasiconformal maps, Monatsch. Math. (to appear).

    Google Scholar 

References

  1. Brown J. E., Geometric properties of a class of support points of univalent functions, Trans. Amer. Math. Soc. 256 (1979), 371–382.

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown J. E., Univalent functions maximizing Re{a 3a 2}, Illinois J. Math. 25 (1981), 446–454.

    MathSciNet  Google Scholar 

  3. Duren P. L., Arcs omitted by support points of univalent functions, Comment. Math. Helv. 56 (1981), 352–365.

    Article  MathSciNet  MATH  Google Scholar 

  4. Duren P. L., Univalent Functions, Springer-Verlag, New York, 1983.

    MATH  Google Scholar 

  5. Hamilton D. H., On Littlewood’s conjecture for univalent functions, Proc. Amer. Math. Soc. 86 (1982) 32–36.

    MathSciNet  MATH  Google Scholar 

  6. Pearce K., New support points of S and extreme points of HS, Proc. Amer. Math. Soc. 81 (1981), 425–428.

    MathSciNet  MATH  Google Scholar 

References

  1. Hayman W. K., Wu J.-M. G., Level sets of univalent functions, Comm. Math. Helv. 56 (1981), 366–403.

    Article  MathSciNet  MATH  Google Scholar 

  2. Garnett J. B., Gehring F. W., Jones P. W. Conformally invariant length sums, Indiana Univ. Math. J. 32 (1983), 809–829.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Victor P. Havin Nikolai K. Nikolski

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag

About this chapter

Cite this chapter

Duren, P. (1994). Geometric function theory. In: Havin, V.P., Nikolski, N.K. (eds) Linear and Complex Analysis Problem Book 3. Lecture Notes in Mathematics, vol 1574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101068

Download citation

  • DOI: https://doi.org/10.1007/BFb0101068

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57871-0

  • Online ISBN: 978-3-540-48368-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics