Let Pn denote the linear space of polynomials p(z:=Σk=0nak(p)zk of degree ≦ n with complex coefficients and let |p|[−1,1]: = maxx∈[−1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [−1, 1]. Let tn ∈ Pn be the nth Chebyshev polynomial. The inequality \documentclass[12pt]{minimal}
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\frac{{\left| p \right|_{\left[ { - 1,1} \right]} }}
{{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }}
{{\left| {a_n (t_n )} \right|}},p \in P_n
$$\end{document} due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial tn in Pn. The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.
