References
L. V. Ahlfors,Conformal Invariants, McGraw-Hill, New York, 1973.
G. M. Golusin,Geometric theory of functions of a complex variable, Trans. Math. Monographs, Vol. 26, Amer. Math. Soc., Providence, R.I., 1969.
J. A. Hummel,A variational method for starlike functions, Proc. Amer. Math. Soc.9 (1958), 82–87.
J. A. Hummel and M. M. Schiffer,Variational methods for Bieberbach-Eilenberg functions and for Pairs, to appear in Ann. Acad. Sci. Fenn.
F. Marty,Sur le module des coefficients de MacLaurin d'une fonction univalent, C. R. Acad. Sci. Paris198 (1934), 1569–1571.
E. Netanyahu,On univalent functions in the unit disc whose image contains a given disc, J. Analyse Math.23 (1970), 305–322.
G. Piranian,An isolated schlicht function, Abh. Math. Sem. Univ. Hamburg24 (1960), 236–238.
Chr. Pommerenke,Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975.
M. S. Robertson,A variational method for functions with positive real part, Trans. Amer. Math. Soc.102 (1962), 82–93.
A. C. Schaeffer and D. C. Spencer,Coefficient regions for schlicht functions, Amer. Math. Soc. Colloquium Publ., Vol. 35, Amer. Math. Soc., New York, 1950.
A. C. Schaeffer, M. Schiffer, and D. C. Spencer,The coefficient regions of schlicht functions, Duke Math. J.16 (1949), 493–526.
M. Schiffer,A method of variations within the family of simple functions, Proc. London Math. Soc. (2)44 (1938), 432–449.
M. Schiffer,Extremal problems and variational methods in conformal mapping, Proc. International Congress Math., Edinburgh (1958), 213–231.
G. Schober,Univalent Functions—Selected Topics, Lecture Notes in Math. 478, Springer-Verlag, Berlin, 1975.
V. Singh,Interior variations and some extremal problems for certain classes of univalent functions, Pacific J. Math.7 (1957), 1485–1504.
Author information
Authors and Affiliations
Additional information
This research was supported in part by the Samuel Neaman Fund, Special Year in Complex Analysis, Technion—ITT, Haifa, Israel and in part by the National Science Foundation grant number MCS 75-07387A01 to the University of Maryland, College Park, Maryland.
Rights and permissions
About this article
Cite this article
Hummel, J.A. Lagrange multipliers in variational methods for univalent functions. J. Anal. Math. 32, 222–234 (1977). https://doi.org/10.1007/BF02803581
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02803581