The Wirtinger–Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the $L_p$-mean ($p\ge0$) of a $2\pi$-periodic function with zero mean value and the $L_q$-norm ($q\ge1$) of the positive cutoff of its derivative. We obtain estimates of the constant from below for $0\le p\le\infty$ and from above for $1\le p\le\infty$ for an arbitrary $1\le q\le\infty$. We write out the values of the sharp constant in the cases $p=2$, $1\le q\le\infty$ and $p=\infty$, $1\le q\le\infty$.