2025/09/26 by Dmitriy Dmitrishin, Dmitrishin, Dmitriy, Alexander M. Stokolos +3
Mathematics · #Holomorphic and Operator Theory #Mathematical functions and polynomials #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.2509.22238
In the class of normalized sine-polynomials S(t), non-negative on [0,π], W.Rogosinski and G.Szegő 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient a3. Their proof is based on the Lukács representation of non-negative algebraic polynomials. This method does not lead to the construction of polynomials attaining the extreme values. We consider the corresponding problem in the framework of normalized typically real polynomials P(z) on the unit disc in \mathbb C. By L.Fejér's method with the additional use of the Chebyshev polynomials of the second kind and their derivatives, we regain the sharp upper and lower estimates for a3 and identify the extremal polynomials. The corresponding statements for sine polynomials follow by the observation S(t)=Im\P(eit)\. For odd N the extremizers are unique, for even N there is a one-parameter family of extremizers.