Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that $$ \lvert u\rvert_s = \left(\sum_{k\in\mathbb Z} \lvert k\rvert^{2s}\lvert \hat u_k\rvert^2\right)^{\!1/2}<\infty $$ Then, for $\,u\in \overset{\circ\,\,\,\,\,}{H^{n+1}}(\mathbb T)$, $n$ positive integer, we have that $$ \left|\int_{-\pi}^\pi u^{(n)}(x)\,\big(u(x)u'(x)\big)^{(n)}\,dx\right|\le c\lvert u\rvert_{L^\infty}\lvert u\rvert_{n+1/2}^2, \tag{1} $$ for some $c$ depending on $n$ and not on $u$. I managed to obtain that $c\le 2^{n+1}$, which is extreme crude!

Question. Is it possible to show that $c=\mathcal O(n)$?

My crude estimate is based on the fact that $\lvert u^2\rvert_s\le 2^{s+1}\lvert u\rvert_{L^\infty}\lvert u\rvert_s $, which can not improve significantly (perhaps by at most a factor of $2$).