We study the sharp inequality between the uniform norm and Lp(0,π/2)-norm of polynomials in the system C = {cos(2k+1)x}∞k=0 of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order n of polynomials as n → ∞ and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials. © 2022, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.