# Schwarzian derivative

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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 [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 in $D$, and it is impossible in this case to increase the constant 2.

#### References

 [1] H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890) [2] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) [3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)