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I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that $P_n(0)=1$ and $$ \max_{z\in\Omega}|P_n(z)|\to \min.$$

Any help and references are strongly appreciated.

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    The [maximum modulus principle](http://mathworld.wolfram.com/MaximumModulusPrinciple.html) would help a great deal here...2012-02-08
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    Hm... Clearly, we can replace maximum over domain $\Omega$ by maximum over boundary $\partial\Omega$. What next?2012-02-08

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