Löwner method

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 [1]. 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

and

in the class of functions , , , that are regular and univalent in .

The Löwner method has been used (see [3]) 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 [3]); 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 [3] 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 [4]).

References

[1]	K. Löwner, "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I" Math. Ann. , 89 (1923) pp. 103–121
[2]	E. Peschl, "Zur Theorie der schlichten Funktionen" J. Reine Angew. Math. , 176 (1936) pp. 61–94
[3]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4]	I.A. Aleksandrov, "Parametric extensions in the theory of univalent functions" , Moscow (1976) (In Russian)

Comments

The Löwner equation has been used to solve the Bieberbach conjecture, [a1]; cf. [a2]. Further references on the method include [a3]–[a6].

References

[a1]	L. de Branges, "A proof of the Bieberbach conjecture" Acta. Math. , 154 (1985) pp. 137–152
[a2]	C.H. FitzGerald, C. Pommerenke, "The de Branges theorem on univalent functions" Trans. Amer. Math. Soc. , 290 (1985) pp. 683–690
[a3]	W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958)
[a4]	C. Pommerenke, "Univalent functions" , Vandenhoeck & Ruprecht (1975)
[a5]	P.L. Duren, "Univalent functions" , Springer (1983) pp. 258
[a6]	D.A. Brannan, "The Löwner differential equation" D.A. Brannan (ed.) J.G. Clunie (ed.) , Aspects of Contemporary Complex Analysis , Acad. Press (1980) pp. 79–95