On the Szego — Taikov inequality for conjugate trigonometric polynomials

In 1943, Szego found the best constant and all extremal polynomials in the inequality between the uniform norm of a conjugate trigonometric polynomial and the norm of the polynomial with real coefficients. In 1990, Taikov also found the best constant by means of another method. In the present paper, we describe all extremal polynomials with complex coefficients and discuss some properties of the extremal polynomials.