# 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 $Q _ {t}$, $0 \leq t < T$, included in one another, $Q _ {t ^ \prime } \subset Q _ {t}$, $0 \leq t ^ \prime < t < T$. For $Q _ {0}$ one assumes known a conformal mapping $f _ {0}$ onto some canonical domain $B _ {0}$. From a known mapping $f _ {t}$ of $Q _ {t}$ onto a domain of canonical form one constructs such a mapping $f _ {t+ \epsilon }$ for $Q _ {t+ \epsilon }$, where $\epsilon > 0$ is small. Under a continuous change of the parameter $t$ 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 $Q _ {t}$ and a natural number $t$— the transition from $f _ {t}$ to $f _ {t + \epsilon }$, $\epsilon = 1$, is effected by recurrence formulas. The source of these formulas and equations is usually the Schwarz formula (see ) and its generalizations (see ). An equally important source of parametric representations are the Hadamard variations (see , ) for the Green functions $G _ {t} ( z, z ^ \prime )$, $z, z ^ \prime \in Q _ {t}$, of the family of domains mentioned above. For elliptic differential equations Hadamard's method is also called the method of invariant imbedding (see ). Below, the connection between parametric representations, Hadamard variations and invariant imbedding are exhibited in the simplest (discrete) case.

Suppose that $Q$ is a collection of complex integers (a lattice domain) and that the Green function $G _ {z} ( z, z ^ \prime )$ is an extremal of the Dirichlet–Douglas functional

$$I _ {t} ( g) = 2g( z ^ \prime ) + \sum _ { k= } 0 ^ { l } \sum _ {z \in Q _ {0} } \rho _ {k} ( t) | \nabla _ {k} g( z) | ^ {2}$$

in the class $R _ {0}$ of all real-valued functions $g( z)$ on $Q$. Here $Q _ {0} = \{ {z } : {z, z- 1, z- i, z- 1- i \in Q } \}$,

$$\tag{1 } \left . \begin{array}{c} {\nabla _ {0} g( z) = g( z) - g( z- 1- i), } \\ {\nabla _ {1} g( z) = g( z- 1)- g( z- i), } \end{array} \right \}$$

$$\rho _ {k} ( 0) \equiv 1,\ \rho _ {k} ( t+ 1) = \rho _ {k} ( t) + N \delta _ {\zeta _ {t} } ,$$

$N$ is a natural number, $\delta _ {\zeta _ {t} }$ is the Kronecker symbol, and $\zeta _ {t} = ( k _ {t} , z _ {t} )$, $t = 0 \dots T- 1$, is a certain collection of pairs of numbers; $\{ {z _ {t} } : {t = 1 \dots T } \}$ is the boundary of $Q _ {t}$, and $k _ {t} = 0$ or 1. To find an extremum of the functional $I _ {t} ( g)$ is a problem of quadratic programming. A comparison of its solutions for $t$ and $t+ 1$ gives the basic formula of invariant imbedding (Hadamard variation):

$$\tag{2 } G _ {t+} 1 ( z, z ^ \prime ) = G _ {t} ( z, z ^ \prime ) - \frac{1}{c _ {t} } \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z) \nabla _ {k _ {t} } G _ {t} ( z _ {t} , z ^ \prime ),$$

where $c _ {t} = N ^ {-} 1 - \nabla _ {k _ {t} } \nabla _ {k _ {t} } ^ \prime G _ {t} ( z _ {t} , z _ {t} ) > 0$ and the symbol $\nabla _ {k} ^ \prime$ denotes the difference operators (1) in the second argument of the Green function. Knowing $G _ {0} ( z, z ^ \prime )$ one can obtain step-by-step (recurrently) from (2) all the functions $G _ {t} ( z, z ^ \prime )$, $t = 1 \dots T$. By constructing the Green function, one obtains from the lattice analytic function $f _ {T} ( z) = G _ {T} ( z, z ^ \prime ) + iH _ {T} ( z, z ^ \prime )$ according to the equation of Cauchy–Riemann type

$$(- 1) ^ {k} \nabla _ {1-} k H = \rho _ {k} \nabla _ {k} G,$$

a univalent lattice quasi-conformal mapping $w = \mathop{\rm exp} [ 2 \pi f( z)]$ of $Q _ {t}$ into the unit disc. Closest to the origin of coordinates is the image of $z ^ \prime$. In the limit, as $n \rightarrow \infty$, the mapping is lattice conformal and the image of $Q _ {T}$ is a disc with concentric slits. The result is a continuous analogue of (2) (see ). When all the domains $G _ {t}$ are simply connected and the canonical domain is the unit disc $B$, one succeeds by using a fractional-linear automorphism of $B$ to represent the Green function in the explicit form

$$G _ {t} ( z, z ^ \prime ) = \mathop{\rm ln} | 1- f _ {t} ( z) \overline{ {f _ {t} ( z ^ \prime ) }}\; | - \mathop{\rm ln} | f _ {t} ( z) - f _ {t} ( z ^ \prime ) |$$

in terms of the function $f _ {t} ( z)$ mapping $Q _ {t}$ onto $B$ with the normalization $f( 0) = 0$, $0 \in Q _ {t}$ for all $t \in [ 0, T)$.

In terms of $w = f _ {t} ( z)$ 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 $Q _ {t}$ is only implicit in it — in terms of the control parameter $\alpha ( t) = \mathop{\rm arg} f _ {t} ( z _ {t} )$, because $f _ {t} ( z _ {t} )$ is not known beforehand. Nevertheless, the Löwner equation is a basic instrument in the parametric representation.

More general one-parameter families of domains $Q _ {t}$, $0 \leq t < T$, not necessarily imbedded in one another, have also been treated (see ). 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 $Q _ {t}$ have a different kind of symmetry or other geometric peculiarities (see ).

How to Cite This Entry:
Parametric representation of univalent functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parametric_representation_of_univalent_functions&oldid=48127
This article was adapted from an original article by V.I. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article