On Extrapolation of Polynomials with Real Coefficients to the Complex Plane

The problem of the greatest possible absolute value of the $k$th derivative of an algebraic polynomial of order $n>k$ with real coefficients at a given point of the complex plane is considered. It is assumed that the polynomial is bounded by $1$ on the interval $[-1,1]$. It is shown that the solution is attained for the polynomial $\kappa\cdot T_\sigma$, where $T_\sigma$ is one of the Zolotarev or Chebyshev polynomials and $\kappa$ is a number.