# Schwarzian derivative

Schwarz derivative, Schwarzian differential parameter, of an analytic function $f(z)$ of a complex variable $z$

The differential expression

$$\{f,z\}=\frac{f'''(z)}{f'(z)}-\frac32\left(\frac{f''(z)}{f'(z)}\right)^2=\left(\frac{f''(z)}{f'(z)}\right)'-\frac12\left(\frac{f''(z)}{f'(z)}\right)^2.$$

It first appeared in studies on conformal mapping of polygons onto the disc, in particular in the studies of H.A. Schwarz .

The most important property of the Schwarzian derivative is its invariance under fractional-linear transformations (Möbius transformations) of the function $f(z)$, i.e. if

$$g(z)=\frac{af(z)+b}{cf(z)+d},$$

then $\{f,z\}=\{g,z\}$. Applications of the Schwarzian derivative are especially connected with problems on univalent analytic functions. For example, if $f(z)$ is a univalent analytic function in the disc $D=\{z:|z|<1\}$, and if $f(0)=0$, $f'(0)=1$, then

$$|\{f,z\}|\leq\frac6{{(1-|z|^2)}^2},\qquad|z|<1.$$

Conversely, if $f(z)$ is regular in $D$ and if

$$|\{f,z\}|\leq\frac2{{(1-|z|^2)}^2},\qquad|z|<1,$$

then $f(z)$ is a univalent function in $D$, and it is impossible in this case to increase the constant 2.

How to Cite This Entry:
Schwarzian derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarzian_derivative&oldid=43567
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article