An estimate of the geometric mean of the derivative of a polynomial in terms of its uniform norm on a closed interval

We study an estimate of the geometric mean of the derivative of an algebraic polynomial of degree at most $n$ in terms of its uniform norm on a closed interval. In the general case, we obtain close two-sided estimates for the best constant; the estimates describe the order growth of the constant with respect to $n$. In the case $n=2$, the best constant is found exactly.