Coefficients of trigonometric polynomials under a one-sided constraint

For odd trigonometric polynomials bounded from above by the function $\varphi(x)=x$ on the intervals $[0, \pi]$ and $[0, 2 \pi]$, we study the maximum and minimum values of coefficients. We obtain two-sided estimates for the first coefficient and find the asymptotic behavior of its maximum and minimum values with respect to the order of the polynomial. We estimate the leading coefficients.