Bernstein-type inequalities

It is shown that a Bernstein-type inequality always implies its Szegő-variant, and several corollaries are derived. Then, it is proven that the original Bernstein inequality on derivatives of trigonometric polynomials implies both Videnskii’s inequality (which estimates the derivative of trigonometric polynomials on a subinterval of the period), as well as its half-integer variant. The methods for these two results are then combined to derive the general sharp form of Videnskii’s inequality on symmetric E⊂[−π,π]E⊂[−π,π] sets. The sharp Bernstein factor turns out to be 2π2π times the equilibrium density of the set ΓE={eit|t∈E}ΓE={eit|t∈E} on the unit circle C1C1 that corresponds to EE when we identify C1C1 by R/(mod2π).