# Schwarzian derivative

*Schwarz derivative, Schwarzian differential parameter, of an analytic function of a complex variable *

The differential expression

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 , i.e. if

then . Applications of the Schwarzian derivative are especially connected with problems on univalent analytic functions. For example, if is a univalent analytic function in the disc , and if , , then

Conversely, if is regular in and if

then is a univalent function in , 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. E.D. Solomentsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Schwarzian_derivative&oldid=14479