Sharp inequalities for trigonometric polynomials with respect to integral functionals

The problem on sharp inequalities for linear operators on the set of trigonometric polynomials with respect to integral functionals $\int_0^{2\pi}\varphi(|f(x)|)\,dx$ is discussed. A solution of the problem on trigonometric polynomials with given leading harmonic that deviate the least from zero with respect to such functionals over the set of all functions $\varphi$ determined, nonnegative, and nondecreasing on the semi-axis $[0,+\infty)$ is given.