Minimal area problems for functions with integral representation
We study the minimization problem for the Dirichlet integral in some standard classes of analytic functions. In particular, we solve the minimal areaa 2-problem for convex functions and for typically real functions. The latter gives a new solution to the minimal areaa 2-problem for the classS of normalized univalent functions in the unit disc.
KeywordsUnivalent Function Conformal Mapping Bergman Space Minimal Area Extremal Function
- [ASh1] D. Aharonov and H. Shapiro,A minimal-area problem in conformal mapping, inProceedings of the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973), Cambridge Univ. Press, London, 1974, pp. 1–5.Google Scholar
- [ASh3] D. Aharonov and H. Shapiro,A minimal area problem in conformal mapping—Preliminary Report, II, Royal Institute of Technology Reseach Report, Stockholm 1978, TRITA-MAT-1978-5.Google Scholar
- [D] P. Duren,Univalent Functions, Springer-Verlag, New York, 1980.Google Scholar
- [HM] D. J. Hallcnbeck and T. H. MacGregor,Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman, Boston, 1984.Google Scholar
- [Sh] H. S. Shapiro,The Schwarz Function and Its Generalization to Higher Dimenstons, Wiley Interscience, New York, 1992.Google Scholar