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Extremal polynomials which approximate a function that conformally maps a given simply-connected domain onto a disc. These polynomials were first studied by L. Bieberbach [[#References|[1]]] in the context of the problem of the approximate computation of a conformal mapping.

Let  $  G $
be a simply-connected domain in the finite part of the plane bounded by a curve  $  \Gamma $, 
and let the function  $  w = \phi (z) $
map this domain conformally and univalently onto the disc  $  | w | < r _ {0} $
under the conditions  $  \phi (z _ {0} ) = 0 $
and  $  \phi  ^  \prime  (z _ {0} ) = 1 $, 
where  $  z _ {0} $
is an arbitrary fixed point of  $  G $
and  $  r _ {0} $
depends on  $  z _ {0} $. 
The polynomial  $  \pi _ {n} (z) $
which minimizes the integral

$$ 
J (F _ {n} )  = \ 
{\int\limits \int\limits } _ {G} | F _ {n} ^ { \prime } (z) |  ^ {2}  dx  dy
$$

in the class of all polynomials  $  F _ {n} (z) $
of degree  $  n $
subject to the conditions  $  F _ {n} (z _ {0} ) = 0 $
and  $  F _ {n} ^ { \prime } (z _ {0} ) = 1 $
is called the Bieberbach polynomial. In the class of all functions which are analytic in the domain  $  G $
and which satisfy the same conditions, this integral is minimized by the mapping function  $  w = \phi (z) $. 
If the contour  $  \Gamma $
is a Jordan curve, the sequence  $  \{ \pi _ {n} (z) \} $
converges uniformly to the function  $  \phi (z) $
inside  $  G $. 
In the closed domain  $  \overline{G}\; $
there need not be convergence [[#References|[2]]]. If the contour  $  \Gamma $
satisfies certain additional smoothness conditions, the sequence  $  \{ \pi _ {n} (z) \} $
converges uniformly in the closed domain, and the rate of convergence depends on the degree of smoothness of  $  \Gamma $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,   "Zur Theorie und Praxis der konformen Abbildung"  ''Rend. Circ. Mat. Palermo'' , '''38'''  (1914)  pp. 98–112</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Keldysh,   "Sur l'approximation en moyenne quadratique des fonctions analytiques"  ''Mat. Sb.'' , '''5 (47)''' :  2  (1939)  pp. 391–401</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.N. Mergelyan,   "Some questions of the constructive theory of functions"  ''Trudy Mat. Inst. Steklov.'' , '''37''' , Moscow  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. Suetin,   "Polynomials orthogonal over a region and Bieberbach polynomials"  ''Proc. Steklov Inst. Math.'' , '''100'''  (1974)  ''Trudy Mat. Inst. Steklov.'' , '''100'''  (1971)</TD></TR></table>

====Comments====
A good additional reference is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Vorlesungen über Approximation im Komplexen" , Birkhäuser  (1980)</TD></TR></table>