Accession Number : AD0282264

Title :   THE THEORY OF THE SECOND VARIATION IN EXTREMUM PROBLEMS FOR UNIVALENT FUNCTIONS,

Corporate Author : STANFORD UNIV CALIF APPLIED MATHEMATICS AND STATISTICS LABS

Personal Author(s) : Duren ,P. L. ; Schiffer,M.

Report Date : 23 JUL 1962

Pagination or Media Count : 84

Abstract : An attempt is made to improve the method of variations by considering further necessary conditions for the extremum function which arises from a study of the second variation. The formulas for the second variation of coefficients of univalent functions are, in general, so involved as to be impractical for a finer study of the extremum function. However, using the fact that this function satisfies a differential equation due to the first variational condition, success was reached in simplifying the expressions considerably. It was shown that a whole new set of necessary extremum conditions can be obtained to test every competing solution of the first variational condition. The new extremum conditions have the form of quadratic inequalities which are similar in type to those occurring through the method of contour integration. The characteristic difference lies in the fact that the quadratic inequalities have only to hold in the case of the extremum function, while in the other case the inequalities are asserted for all univalent functions. Nevertheless, it seems that the theory of the second variation is more closely connected with the method of contour integration than is that of the first variation. It might be possible to combine both methods for a unified approach to the general coefficient problem. (Author)

Descriptors :   *CALCULUS OF VARIATIONS, *DIFFERENTIAL EQUATIONS, *FUNCTIONS(MATHEMATICS), GREEN'S FUNCTION, INEQUALITIES, NUMERICAL INTEGRATION

Distribution Statement : APPROVED FOR PUBLIC RELEASE