Markov Factors on Average -- an L_infty Case

The classical Markov polynomial inequality bounds the norm of the derivative of a polynomial on an interval in terms of its degree squared and the norm of the polynomial itself with the factor of degree squared being the optimal (worst case) upper bound. Here we study what this factor should be on average, for random polynomials with independent, identically distributed, positive random coefficients. In the case of the interval [0, t], t > 0, we show that this average factor is of order the degree for t >= 1 while for t < 1, it is a constant, independent of the degree. We give some numerical experiments that indicate that the same behaviour holds for when the coefficients are allowed to be negative, e.g., independent N(0, 1) random variables, as well as for the symmetric intervals [-t, t].