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The theory of the second variation in extremum problems for univalent functions

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References

  1. L. Bieberbach, Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln,Preuss. Akad. Wiss., Sitzungsberichte, 1916, pp. 940–955.

    Google Scholar 

  2. Z. Charzynski and M. Schiffer, A new proof of the Bieberbach conjecture for the fourth coefficient,Archive Rat. Mech. Anal., vol. 5 (1960), pp. 187–193.

    Article  MATH  MathSciNet  Google Scholar 

  3. Z. Charzynski and M. Schiffer, A geometric proof of the Bieberbach conjecture for the fourth coefficient,Scripta Mathematica, vol. 25 (1960), pp. 173–181.

    MATH  MathSciNet  Google Scholar 

  4. R. Courant, Dirichlet's Principle, Conformal mapping, and Minimal Surfaces. New York, 1950. Appendix by M. Schiffer.

  5. P. R. Garabedian and M. Schiffer, Convexity of domain functionals,Jour. d'Analyse Math., vol. 2 (1952–53), pp. 281–368.

    Article  MathSciNet  Google Scholar 

  6. P. R. Garabedian and M. Schiffer, A coefficlent inequality for schlicht functions,Ann. Math., vol. 61 (1955), pp. 116–136.

    Article  MathSciNet  Google Scholar 

  7. P. R. Garabedian and M. Schiffer, A proof of the Bieberbach conjecture for the fourth coefficient,Jour. Rat. Mech. Anal., vol. 4 (1955), pp. 427–465.

    MathSciNet  Google Scholar 

  8. G. M. Golusin, Geometrical Theory of Functions of a Complex Variable. Moscow, 1955: German translation: Berlin, 1957.

  9. H. Grötzsch, Über einige Extremalprobleme der konformen Abbildung,Leipzig Ber, vol. 80 (1928), pp. 367–376.

    Google Scholar 

  10. H. Grunsky, Koeffizientenbedingungen für schlicht abbildende Funktionen,Math. Zeitschr., vol. 45 (1939), pp. 29–61.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Hadamard, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées,Mémoires présentés par divers savants à l'Académie des Sciences, vol. 33 (1908).

  12. J. Hadamard, Lećcons sur le calcul des variations. Paris, 1910.

  13. J. A. Jenkins, Univalent Functions and Conformal Mapping. Springer-Verlag, 1958.

  14. P. Lévy, Problèmes concrets d'analyse fonctionelle. Paris, 1951. (Second edition of Leçons d'analyse fonctionelle, Paris, 1922).

  15. K. Loewner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises I,Math. Ann., vol. 89 (1923), pp. 103–121.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Marty, Sur le module des coefficients de MacLaurin d'une fonction univalente,Comptes Rendus Acad. Sci., Paris vol. 198 (1934), pp. 1569–1571.

    MATH  Google Scholar 

  17. P. Montel, Leçons sur les fonctions univalentes ou multivalentes. Paris, 1933.

  18. A. C. Schaeffer and D. C. Spencer, The coefficients of schlicht functions I,Duke Math. Jour., vol. 10 (1943), pp. 611–635; II,Duke Math. Jour., vol. 12 (1945), pp. 107–125; III,Proc. Nat. Acad. Sci., Wash., vol. 32 (1946), pp. 111–116.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. C. Schaeffer and D. C. Spencer, A variational method in conformal mapping,Duke Math. Jour., vol. 14 (1947), pp. 949–966.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht Functions.Amer. Math. Soc. Colloq. Publ., No. 35, 1950.

  21. A. C. Schaeffer, M. Schiffer and D. C. Spencer, The coefficient regions of schlicht functions,Duke. Math. Jour., vol. 16 (1949), pp. 493–527.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Schiffer, A method of variation within the family of simple functions,Proc. London Math. Soc., vol. 44 (1938), pp. 432–449.

    Article  MATH  Google Scholar 

  23. M. Schiffer, On the coefficients of simple functions,Proc. London Math. Soc., vol. 44 (1938), pp. 450–452.

    Article  MATH  Google Scholar 

  24. M. Schiffer, Variation of the Green's function and theory of thep-valued functions,Amer. Jour. Math., vol. 65 (1943), pp. 341–360.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Schiffer, Hadamard's formula and variation of domain-functions,Amer. Jour. Math., vol. 68 (1946), pp. 417–448.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Schiffer, Faber polynomials in the theory of univalent functions,Bull. Amer. Math. Soc., vol. 54 (1948), pp. 503–517.

    MATH  MathSciNet  Google Scholar 

  27. M. Schiffer, Application of variational methods in the theory of conformal mappings,Proc. Symp. Appl. Math., vol. 8 (1958), pp. 93–113.

    MathSciNet  Google Scholar 

  28. M. Schiffer, Extremum problems and variational methods in conformal mapping,Proceedings of the International Congress of Mathematicians, 1958.

  29. M. Schiffer and D. C. Spencer, Functionals of Finite Riemann Surfaces, Princeton, 1954.

  30. O. Teichmüller, Ungleichungen zwischen den Koeffizienten schlichter Funktionen,Preuss. Akad. Wiss., Sitzungsherichte, 1938, pp. 363–375.

    Google Scholar 

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This work was supported in part by Office of Naval Research Contract Nonr—225 (11) at Stanford University. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Duren, P.L., Schiffer, M. The theory of the second variation in extremum problems for univalent functions. J. Anal. Math. 10, 193–252 (1962). https://doi.org/10.1007/BF02790308

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