View source for Schwarzian derivative

{{TEX|done}}
''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|conformal mapping]] of polygons onto the disc, in particular in the studies of H.A. Schwarz [[#References|[1]]].

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|univalent function]] in $D$, and it is impossible in this case to increase the constant 2.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.A. Schwarz,   "Gesamm. math. Abhandl." , '''2''' , Springer  (1890)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>



====Comments====
The necessary and sufficient conditions for univalency in terms of the Schwarzian derivative stated above are due to W. Kraus [[#References|[a1]]] and Z. Nehari [[#References|[a2]]], respectively; see [[#References|[a3]]], pp. 258-265, for further discussion. A nice discussion of the Schwarzian derivative is in [[#References|[a4]]], pp. 50-58.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Kraus,   "Ueber den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung"  ''Mitt. Math. Sem. Giessen'' , '''21'''  (1932)  pp. 1–28</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Z. Nehari,   "The Schwarzian derivative and schlicht functions"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 545–551</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. 258</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O. Lehto,   "Univalent functions and Teichmüller spaces" , Springer  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Z. Nehari,   "Conformal mapping" , Dover, reprint  (1975)  pp. 2</TD></TR></table>