The Isoperimetric Inequality and Rational Approximation

REFERENCES

• L. Ahlfors and A. Beurling, Conformal invariants and function theoretic null sets, Acta Math., 83 (1950) 101–129.
   Web of Science ®Google Scholar
• H. Alexander, Projections of polynomial hulls, J. Fund. Anal., 3 (1973) 13–19.
   Google Scholar
• H. Alexander, On the area of the spectrum of an element of a uniform algebra, Complex Approximation, Proceedings, Quebec, July 3–8, 1978, ed. by B. Aupetit, Birkhauser, 1980, pp. 3–12.
   Google Scholar
• S. Axler and J. Shapiro, Putnam's theorem, Alexander's spectral area estimate, and BMO, Math. Ann., 271 (1985) 161–183.
   Web of Science ®Google Scholar
• C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston-London-Melbourne, 1980.
   Google Scholar
• W. Blaschke, Kreis und Kugel, Second Edition, deGruyter, Berlin, 1956.
   Google Scholar
• A. Browder, Introduction to Function Algebras, Benjamin, New York, 1969.
   Google Scholar
• C. Carathéodory and E. Study, Zwei Beweise des Satzes, dass der Kreis unter allen Figuren gleichen Umfanges den grössten Inhalt hat, Math. Ann., 58 (1910) 133–140.
   Google Scholar
• T. Carleman, Zur Theorie der Minimalflächen, Math. Zeit., 9 (1921) 154–160.
   Google Scholar
• F. Edler, Vervollständigung der Steinersche elementargeometrischen Beweise für den Satz, dass der Kreis grösseren Flächeninhalt besitzt als jede andere Figur gleich grossen Umfanges, Gött. Nachr., 1882, p. 73.
   Google Scholar
• H. Federer, Geometrie Measure Theory, Springer-Verlag, Heidelberg, New York, 1969.
   Google Scholar
• B. Fuglede, Stability in the isoperimetric problem, Matematisk Institut, Copenhagen (1986), preprint.
   Google Scholar
• T. Gamelin, Uniform Algebras, Second Edition, Chelsea Press, 1984.
   Google Scholar
• T. Gamelin, On an estimate of Axler and Shapiro, Math. Ann., 272 (1985) 189–196.
   Web of Science ®Google Scholar
• A. Hurwitz, Sur le problème des isopérimètres, C. R. Acad. Sci. Paris, 132 (1901) 401–403.
   Google Scholar
• S. Jacobs, An isoperimetric inequality for functions analytic in multiply connected domains, Ph.D. Thesis, Institut Mittag-Leffler, 1967.
   Google Scholar
• D. Khavinson, Sets of finite perimeter, Cauchy integrals and rational approximation, Approximation Theory IV, ed. by C. K. Chui, L. L. Schumacher and J. D. Ward, Academic Press, New York, 1983, pp. 567–574.
   Google Scholar
• D. Khavinson, Annihilating measures of the algebra R(X), J. Fund. Anal., 58 (1984) 175–193.
   Web of Science ®Google Scholar
• D. Khavinson, A note on Toeplitz operators, Proceedings of the NSF-CBMS regional conference “Geometry of Banach Spaces” at the University of Missouri, Columbia, June 1984, Springer Lecture Notes in Math., Vol. 1166 (1986) 89–95.
   Google Scholar
• D. Khavinson, The Cauchy-Green formula and its applications to problems in rational approximation on sets with a finite perimeter in the complex plane, J. Funct. Anal., 64 (1985) 112–123.
   Web of Science ®Google Scholar
• D. Khavinson and D. Luecking, On an extremal problem in the theory of rational approximation, J. Approximation Theory, 50 (1987) 127–132.
   Web of Science ®Google Scholar
• A. I. Markushevich, Theory of Analytic Functions, volumes I, II and III, English Transl, ed. by R. Silverman, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
   Google Scholar
• S. N. Mergelyan, Uniform approximation to functions of a complex variable, Amer. Math. Soc. Translations, 101 (1954).
   Google Scholar
• S. N. Mergelyan, On a theorem of M. A. Lavrentiev, Amer. Math. Soc. Translations, Ser. 2, 86 (1953) 3–7.
   Google Scholar
• R. Osserman, Isoperimetric inequalities, Bull. Amer. Math. Soc., 84 (1978) 1182–1238.
   Web of Science ®Google Scholar
• G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, N. J., 1951.
   Google Scholar
• S. Saitoh, The Bergman norm and the Szegö norm, Trans. Amer. Math. Soc., 249 (1979) 261–279.
   Web of Science ®Google Scholar
• E. Schmidt, Ueber das isoperimetrische Problem in Raum von n Dimensionen, Math. Z., 44 (1939) 689–788.
   Google Scholar
• H. A. Schwarz, Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumens, Gött. Nachr., 1884, pp. 1–13; also in H. A. Schwatz, Gesammelte Abhandlungen, vol. 2, Springer Verlag, Berlin, 1890, pp. 327–340.
   Google Scholar
• C. Stanton, Counting functions and majorization for Jensen measures, preprint.
   Google Scholar
• D. Stegenga, A geometric condition which implies BMOA, in Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure Math., Vol. 35, Part 1, Amer. Math. Soc., Providence, 1979, pp. 427–430.
   Google Scholar
• J. Steiner, Einfache Beweise der isoperimetrischen Hauptsätze, J. reine ang. Math., 18 (1838) 281–296; also in J. Steiner, Gesammelte Werke, vol. 2, Reimer, Berlin, 1882, pp. 75–91.
   Google Scholar
• A. G. Vitushkin, Analytic capacity of sets and problems in approximation theory, Russian Math. Surveys, 22 (1967) 139–200.
   Google Scholar