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''Schwarz derivative, Schwarzian differential parameter, of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836001.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836002.png" />''
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{{TEX|done}}
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''Schwarz derivative, Schwarzian differential parameter, of an analytic function $f(z)$ of a complex variable $z$''
  
 
The differential expression
 
The differential expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836003.png" /></td> </tr></table>
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$$\{f,z\}=\frac{f'''(z)}{f'(z)}-\frac32\left(\frac{f''(z)}{f'(z)}\right)^2=\left(\frac{f''(z)}{f'(z)}\right)'-\frac12\left(\frac{f''(z)}{f'(z)}\right)^2.$$
  
 
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]]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836004.png" />, i.e. if
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836005.png" /></td> </tr></table>
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$$g(z)=\frac{af(z)+b}{cf(z)+d},$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836006.png" />. Applications of the Schwarzian derivative are especially connected with problems on univalent analytic functions. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836007.png" /> is a univalent analytic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836008.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s0836009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360010.png" />, then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360011.png" /></td> </tr></table>
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$$|\{f,z\}|\leq\frac6{{(1-|z|^2)}^2},\qquad|z|<1.$$
  
Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360012.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360013.png" /> and if
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Conversely, if $f(z)$ is regular in $D$ and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360014.png" /></td> </tr></table>
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$$|\{f,z\}|\leq\frac2{{(1-|z|^2)}^2},\qquad|z|<1,$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360015.png" /> is a [[Univalent function|univalent function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083600/s08360016.png" />, and it is impossible in this case to increase the constant 2.
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then $f(z)$ is a [[Univalent function|univalent function]] in $D$, and it is impossible in this case to increase the constant 2.
  
 
====References====
 
====References====

Latest revision as of 11:10, 30 December 2018

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)


Comments

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

References

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