# Bieberbach polynomials

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 [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 [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

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