Short note on Bernstein's Inequality

The famous Bernstein’s inequality estimates the absolute value of a polynomial's derivative on the unit circle via the maximum absolute value of that polynomial over the circle. In this paper, we prove an explicit formula for increment of a polynomial along a ray, which allows to replace the maximum of absolute value over the unit circle by the maximum through the vertices of an inscribed regular polygon. As a consequence, a new proof of a discrete variant of Bernstein’s polynomial inequality is given.