Proof of a conjecture of Erdős about the longest polynomial

In 1939 P. Erdös conjectured that the Chebyshev polynomial T n ( x ) {T_n}(x) has a maximal arc-length in [ − 1 , 1 ] [ - 1,1] among the polynomials of degree n n which are bounded by 1 in [ − 1 , 1 ] [ - 1,1] . We prove this conjecture for every natural n n .