On one inequality of different metrics for trigonometric polynomials

We study the sharp inequality between the uniform norm and $L^p(0,\pi/2)$-norm of polynomials in the system $\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty$ 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 $n\to\infty$ and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials.