On the maximum area of inscribed polygons

Given a convex $n$-gon $P$ and positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area $m$-gon contained in $P$. We are interested minimum value of $\Delta(Q)/\Delta(P)$, ratio areas these two polygons. More precisely, given integers $n$ $m$, with $3 \le m define \begin{equation*} f_n(m)=\min_{P\in \mathcal {P}_n} \max_{Q \subset P,|Q|=m} \frac{\Delta(Q)}{\Delta(P)} \end{equation*} where maximum is taken over all $m$-gons $P$, $\mathcal{P}_n$, entire class $n$-gons. The values $f_4(3)$, $f_5(4)$ $f_6(3)$ known. In this paper we compute $f_5(3)$, $f_6(5)$ $f_6(4)$. addition, prove for $n\ge 6$ have \frac{4}{n}\cdot\sin^2\left(\frac{\pi}{n}\right)\le 1-f_n(n-1)\le \min\left(\frac{1}{n}, \frac{4}{n}\cdot\sin^2\left(\frac{2\pi}{n}\right)\right). These bounds can be used to improve known estimates $f_n(m)$.