On Two Polynomial Inequalities of Erdős Related to Those of the Brothers Markov

Polynomials of degree at mostnwhich are real on the real axis and do not vanish in the open unit disk are considered. Sharp point-wise bounds for the derivativep′(x) at an arbitrarily prescribed pointx0of the unit interval, in terms of the maximum of |p(x)| on [−1, 1] are obtained. Certain other related problems are also solved.