An extremal problem for trigonometric polynomials

Let $T_{n}(x)=\sum _{k=0}^{n}(a_{k}\cos kx+b_{k}\sin kx)$ be a trigonometric polynomial of degree $n.$ The problem of finding $C_{np},$ the largest value for $C$ in the inequality $\max \{\left| a_{0}\right| ,\left| a_{1}\right| ,...,\left| a_{n}\right| ,\left| b_{1}\right| ,...,\left| b_{n}\right| \}$ $\leq (1/C)\left\| T_{n}\right\| _{p}$ is studied. We find $C_{np}$ exactly provided $p$ is the conjugate of an even integer $2s$ and $n\geq 2s-1,s=1,2,....$ For general $p,1\leq p\leq \infty ,$we get an interval estimate for $C_{np},$ where the interval length tends to $0$ as $n$tends to $\infty .$