Parametric representation of univalent functions

A representation of univalent functions (cf. Univalent function) that effect a conformal mapping of planar domains onto domains of a canonical form (for example, a disc with concentric slits); it usually arises in the following manner. One chooses a one-parameter family of domains , , included in one another, , . For one assumes known a conformal mapping onto some canonical domain . From a known mapping of onto a domain of canonical form one constructs such a mapping for , where is small. Under a continuous change of the parameter there arise in this way differential equations. The best known of these are the Löwner equation and the Löwner–Kufarev equation. In the discrete case — for lattice domains and a natural number — the transition from to , , is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see [1]) and its generalizations (see [2]). An equally important source of parametric representations are the Hadamard variations (see [3], [4]) for the Green functions , , of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see [5]). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.

Suppose that is a collection of complex integers (a lattice domain) and that the Green function is an extremal of the Dirichlet–Douglas functional

in the class of all real-valued functions on . Here ,

(1)

is a natural number, is the Kronecker symbol, and , , is a certain collection of pairs of numbers; is the boundary of , and or 1. To find an extremum of the functional is a problem of quadratic programming. A comparison of its solutions for and gives the basic formula of invariant imbedding (Hadamard variation):

(2)

where and the symbol denotes the difference operators (1) in the second argument of the Green function. Knowing one can obtain step-by-step (recurrently) from (2) all the functions , . By constructing the Green function, one obtains from the lattice analytic function according to the equation of Cauchy–Riemann type

a univalent lattice quasi-conformal mapping of into the unit disc. Closest to the origin of coordinates is the image of . In the limit, as , the mapping is lattice conformal and the image of is a disc with concentric slits. The result is a continuous analogue of (2) (see [6]). When all the domains are simply connected and the canonical domain is the unit disc , one succeeds by using a fractional-linear automorphism of to represent the Green function in the explicit form

in terms of the function mapping onto with the normalization , for all .

In terms of the Hadamard variation reduces to an ordinary (Löwner) differential equation. In comparison with the Hadamard variation this equation is considerably simpler; however, information on the boundary of is only implicit in it — in terms of the control parameter , because is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.

More general one-parameter families of domains , , not necessarily imbedded in one another, have also been treated (see [7]). The equations arising in such parametric representations are called Löwner–Kufarev equations. There is also a modification of the Löwner and Löwner–Kufarev equations to the case when the domains have a different kind of symmetry or other geometric peculiarities (see [1]).

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[2]	I.A. Aleksandrov, A.S. Sorokin, "The problem of Schwarz for multiply connected circular domains" Sib. Math. J. , 13 : 5 (1972) pp. 671–692 Sibirsk. Mat. Zh. , 13 : 5 (1972) pp. 971–1000
[3]	J. Hadamard, "Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées" , Oeuvres , 2 , CNRS (1968) pp. 515–642
[4]	J. Hadamard, "Leçons sur le calcul des variations" , 1 , Hermann (1910)
[5]	R. Bellma, E. Angel, "Dynamic programming and partial differential equations" , Acad. Press (1972)
[6]	V.I. Popov, "Quantization of control systems" Soviet Math. Dokl. , 13 : 6 (1972) pp. 1668–1672 Dokl. Akad. Nauk. SSSR , 207 : 5 (1972) pp. 1048–1050
[7]	P.P. Kufarev, "On one-parameter families of analytic functions" Mat. Sb. , 13 : 1 (1943) pp. 87–118 (In Russian) (English abstract)

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11