Löwner's method of parametric representation of univalent functions, Löwner's parametric method
A method in the theory of univalent functions that consists in using the Löwner equation to solve extremal problems. The method was proposed by K. Löwner . It is based on the fact that the set of functions , , that are regular and univalent in the disc and that map onto domains of type (cf. Smirnov domain), which are obtained from the disc by making a slit along a part of a Jordan arc starting from a point on the circle and not passing through the point , is complete (in the topology of uniform convergence of functions inside ) in the whole family of functions , , that are regular and univalent in and are such that in . Associating the length of the arc that has been removed with a parameter , it has been established that a function , , that maps univalently onto a domain of type is a solution of the differential equation (see Löwner equation)
, satisfying the initial condition . Here and is a continuous complex-valued function on the interval corresponding to with . Löwner used this method to obtain sharp estimates of the coefficients and , in the expansions
in the class of functions , , , that are regular and univalent in .
The Löwner method has been used (see ) to obtain fundamental results in the theory of univalent functions (distortion theorems, reciprocal growth theorems, rotation theorems). Let be the subclass of functions in that have in the representation
where , as a function of , is regular and univalent in , in , , , and as a function of , , is a solution of the differential equation (*) satisfying the initial condition ; in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval . To estimate any quantity on the class it is sufficient to estimate it on the subclass , since any function of class can be approximated by functions , , , each of which maps univalently onto the -plane with a slit along a Jordan arc starting at and not passing through , and hence by functions . Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function .
Löwner's method has been used in work on the theory of univalent functions (see ); it often leads to success in obtaining explicit estimates, but as a rule it does not ensure the classification of all extremal functions and does not give complete information about their uniqueness. For a complete solution of extremal problems Löwner's method is usually combined with a variational method (see  and Variation-parametric method). Löwner's method has been extended to doubly-connected domains. A generalized equation of the type of Löwner's equation has been obtained for multiply-connected domains and for automorphic functions (see ).
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|||E. Peschl, "Zur Theorie der schlichten Funktionen" J. Reine Angew. Math. , 176 (1936) pp. 61–94|
|||G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)|
|||I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)|
|[a1]||L. de Branges, "A proof of the Bieberbach conjecture" Acta. Math. , 154 (1985) pp. 137–152|
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|[a4]||C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)|
|[a5]||P.L. Duren, "Univalent functions" , Springer (1983) pp. 258|
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Löwner method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=L%C3%B6wner_method&oldid=23408