Let be compact and symmetric with respect to the origin and let . Denote by the equilibrium density of on the unit circle . It is proved that if is an inner point of then for any trigonometric polynomial of degree at most we have
This Bernstein-type inequality is sharp. In particular, if with some , then
So for , he gets from (1) the Videnskii inequality. Also, it is shown that the original Bernstein inequality implies its Szegő variant as well as both Videnskii’s inequality and its half-integer variant.