Summary: Let \(t_n(x)\) be any real trigonometric polynomial of degree \(n\) such that \(\| t_n\|_\infty\leq 1\). Here, we are concerned with obtaining the best possible upper estimate of \[ \int^{2\pi}_0 | t^{(k)}_n(x)|^q dx\Biggl/\int^{2\pi}_0| t^{(k)}_n(x)|^{q- 2}dx, \] where \(q>2\). In addition, we obtain the estimate of \(\| t^{(k)}_n\|_q\) in terms of \(\| t_n\|_q\) and \(\| t^{(r)}_n\|_q\).