Refinement of an inequality of P. L. Chebyshev

Let P _n denote the linear space of polynomials p(z:=Σ _k=0 ⁿ a _k (p)z ^k of degree ≦ n with complex coefficients and let |p|_?1,1]: = max _x∈?1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval ?1, 1]. Let t _n ∈ P _n be the n ^th Chebyshev polynomial. The inequality $$ \frac{{\left| p \right|_{\left { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n $$ due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t _n in P _n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.