Estimates of the derivatives of trigonometric polynomials (Q1101633)

Let f,g be two continuous, \(2\pi\)-periodical functions and \(T_ n\) a trigonometrical polynomial, such that \(f(x)\leq T_ n(x)\leq g(x)\). The author establishes some inequalities of Bernstein type; more precisely, he improves a well-known result of Timan proving that, under the above hypotheses about f,g, holds the following inequality: \[ | T_ n'(x)| \leq \frac{g(x)-f(x)}{2}n(1+C(f,g)\sqrt{\omega (f,1/n)+\omega (\quad g,1/n))}, \] where C(f,g) is a constant depending only on f and g and \(\omega\) (f,t) is the modulus of continuity of f. From here, he also obtains the following estimate \[ \lim_{n}\sup_{T_ n}\{| T_ n'(x)| /nr(x);\quad f(x)\leq T_ n(x)\leq g(x)\}\geq 1, \] where \(r(x)=(g(x)-f(x))/2\).