Pointwise Remez inequality

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $$[-1,1]$$ [ - 1 , 1 ] if they are bounded by 1 on a subset of $$[-1,1]$$ [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.