# 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 $f ( z)$, $f ( 0) = 0$, that are regular and univalent in the disc $E = \{ {z } : {| z | < 1 } \}$ and that map $E$ onto domains of type $( s)$( cf. Smirnov domain), which are obtained from the disc $| w | < 1$ by making a slit along a part of a Jordan arc starting from a point on the circle $| w | = 1$ and not passing through the point $w = 0$, is complete (in the topology of uniform convergence of functions inside $E$) in the whole family of functions $f ( z)$, $f ( 0) = 0$, that are regular and univalent in $E$ and are such that $| f ( z) | < 1$ in $E$. Associating the length of the arc that has been removed with a parameter $t$, it has been established that a function $w = f ( z)$, $f ( 0) = 0$, that maps $E$ univalently onto a domain $D$ of type $( s)$ is a solution of the differential equation (see Löwner equation)

$$\tag{* } \frac{\partial f ( z , t ) }{\partial t } = - f ( z , t ) \frac{1 + k ( t) f ( z , t ) }{1 - k ( t) f ( z , t ) } ,$$

$f ( z , t _ {0} ) = f ( z)$, satisfying the initial condition $f ( z , 0 ) = z$. Here $t \in [ 0 , t _ {0} ]$ and $k ( t)$ is a continuous complex-valued function on the interval $[ 0 , t _ {0} ]$ corresponding to $D$ with $| k ( t) | = 1$. Löwner used this method to obtain sharp estimates of the coefficients $c _ {3}$ and $b _ {n}$, $n = 2 , 3 \dots$ in the expansions

$$w = f ( z) = z + \sum _ { n= } 2 ^ \infty c _ {n} z ^ {n}$$

and

$$z = f ^ { - 1 } ( w) = w + \sum _ { n= } 2 ^ \infty b _ {n} w ^ {n}$$

in the class $S$ of functions $w = f ( z)$, $f ( 0) = 0$, $f ^ { \prime } ( 0) = 1$, that are regular and univalent in $E$.

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 $S ^ { \prime }$ be the subclass of functions $f ( z)$ in $S$ that have in $E$ the representation

$$f ( z) = \lim\limits _ {t \rightarrow \infty } \ e ^ {t} f ( z , t ) ,$$

where $f ( z , t )$, as a function of $z$, is regular and univalent in $E$, $| f ( z , t) | < 1$ in $E$, $f ( 0 ,t ) = 0$, $f _ {z} ^ { \prime } ( 0 , t ) > 0$, and as a function of $t$, $0 < t < \infty$, is a solution of the differential equation (*) satisfying the initial condition $f ( z , 0 ) = z$; $k ( t)$ in (*) is any complex-valued function that is piecewise continuous and has modulus 1 on the interval $[ 0 , \infty )$. To estimate any quantity on the class $S$ it is sufficient to estimate it on the subclass $S ^ { \prime }$, since any function $f ( z)$ of class $S$ can be approximated by functions $f _ {n} ( z)$, $f _ {n} ( 0) = 0$, $f _ {n} ^ { \prime } ( 0) > 0$, each of which maps $E$ univalently onto the $w$- plane with a slit along a Jordan arc starting at $\infty$ and not passing through $w = 0$, and hence by functions $f _ {n} ( z) / f _ {n} ^ { \prime } ( 0) \in S ^ { \prime }$. Under this approximation the quantities to be estimated for the approximating functions converge to the same quantity as for the function $f ( z)$.

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)