Schwarz derivative, Schwarzian differential parameter, of an analytic function of a complex variable
The differential expression
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.
|||H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890)|
|||R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)|
|||G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)|
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.
|[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|
Schwarzian derivative. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Schwarzian_derivative&oldid=14479