Abstract
LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU. For 0<r<1 andr(1+r)−2<b<r(1−r)−2, letS(r, b) be the subclass of functionsf∈S such that |f(r)|=b. In Theorem 1, we solve the problem of minimizing the Dirichlet integral inS(r, b). The first main ingredient of the solution is the establishment of sufficient regularity of the domains onto whichU is mapped by extremal functions, and here techniques of symmetrization and polarization play an essential role. The second main ingredient is the identification of all Jordan domains satisfying a certain kind of functional equation (called “quadrature identities”) which are encountered by applying variational techniques. These turn out to be conformal images ofU by mappings of a special form involving a logarithmic function. In Theorem 2, this aspect of our work is generalized to encompass analogous minimal area problem when a larger number of initial data are prescribed.
Similar content being viewed by others
References
[AS1] D. Aharonov and H. S. Shapiro,A minimal area problem in conformal mapping, Preliminary Report, Royal Institute of Technology Research Report, TRITA-MAT-1973-7, Stockholm, 1973.
[AS2] D. Aharonov and H. Shapiro,A minimal area problem in conformal mapping, inCanterbury 1973 (J. Clunie and W. K. Hayman, eds.), London Math. Soc. Lecture Note Ser.12, Cambridge Univ. Press, 1973, pp. 1–5.
[AS3] D. Aharonov and H. S. Shapiro,Domains on which analytic functions satisfy quadrature identities, J. Analyse Math.30 (1976), 39–73.
[AS4] D. Aharonov and H. S. Shapiro,A minimal area problem in conformal mapping, Preliminary Report, II, Royal Institute of Technology Research Report, TRITA-MAT-1978-5, Stockholm, 1978.
[ASS] D. Aharonov, H. S. Shapiro and A. Yu. Solynin,A minimal area problem in conformal mapping, J. Analyse Math.78 (1999), 157–176.
[Ba] A. Baenstein II,Integral means, univalent functions and circular symmetrization, Acta Math.133 (1974), 139–169.
[Be] S. Bergman,The Kernel Function and Conformal Mapping, Math. Surveys Monographs No. 5, Amer. Math. Soc., Providence, RI, 1950.
[Da] P. Davis,The Schwarz Function and Its Applications, Carus. Math. Monographs17, Math. Assoc. America, Washington, DC, 1974.
[Du] V. N. Dubinin,Symmetrization in geometric theory of functions of a complex variable, Uspehi Mat. Nauk49 (1994), 3–76 (in Russian); English translation: Russian Math. Surveys49 (1994), 1–79.
[H] W. K. Hayman,Multivalent Functions, Cambridge Univ. Press, Cambridge, 1958.
[J] J. A. Jenkins,On circularly symmetric functions, Proc. Amer. Math. Soc.6 (1955), 620–624.
[K1] G. V. Kuz'mina,On extremal properties of quadratic differentials with strip domains in the structure of the trajectories, Zap. Nauchn. Sem. LOMI154 (1986), 110–129; English translation: J. Soviet Math.43 (1988), 2579–2591.
[K2] G. V. Kuz'mina,Methods of geometric function theory. II., Algebra i Analiz9 (1997), 1–50; English translation: St. Petersburg Math. J.9 (1998), 889–930.
[N] Z. Nehari,Conformal Mapping, McGraw-Hill, New York, 1952.
[Sa] M. Sakai,Quadrature Domains, Lecture Notes in Math.934, Springer-Verlag, Berlin, 1982.
[Sh] H. S. Shapiro,The Schwarz Function and Its Generalization to Higher Dimensions, Wiley-Interscience, New York, 1992.
[Shi] M. Shiba,The Euclidian, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems, Kodai Math. J.16 (1993), 118–137.
[So1] A. Yu. Solynin,The boundary distortion and extremal problems in certain classes of univalent functions, Zap. Nauchn. Sem. POMI204 (1993), 115–142 (in Russian); English translation: J. Math. Sci.79 (1996), 1341–1358.
[So2] A. Yu. Solynin,Modules and extremal metric problems, Algebra i Analiz11 (1999), no. 1, 3–86 (in Russian); English translation: St. Petersburg Math. J.11 (2000), 1–65.
[W] V. Wolontis,Properties of conformal invariants, Amer. J. Math.74 (1952), 587–606.
Author information
Authors and Affiliations
Corresponding author
Additional information
The third author thanks for its hospitality the Mittag-Leffler Institute of Royal Swedish Academy of Sciences where this work was finalized. This author was supported in part by the Swedish Institute and by the Russian Fund for Fundamental Research, grant no. 97-01-00259.
Rights and permissions
About this article
Cite this article
Aharonov, D., Shapiro, H.S. & Solynin, A.Y. A minimal area problem in conformal mapping II. J. Anal. Math. 83, 259–288 (2001). https://doi.org/10.1007/BF02790264
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02790264