I want to find $M_{n}$ such that
$$ M_{n}=\inf_{p_{n}\in\bar{\prod}_{n}}\max\limits_{x\in[a,b]}|p_{n}(x)|, $$ where $\overline{\prod}\limits_{n}$ is the set of all monic polynomial degree $n$.
I was studying error analysis, and I have to minimize the $M_{n}$ to find the exact error.
For case $n=1$, $\max\limits_{x\in\left[a,b\right]}|p_{1}(x)|=\max\left(|p_{1}\left(a\right)|,\ |p_{1}\left(b\right)|\right)$.
For case $n=2$, (I made short here) If $p_{2}\left(x\right)=x^{2}+mx+n$, $\max\limits_{x\in\left[a,b\right]}|p_{2}(x)|=\max\left(|-\frac{m^{2}}{4}+n|,\ |p_{1}\left(a\right)|,\ |p_{2}\left(b\right)|\right)$. It is difficult for me to find $M_{2}$.
I need help here. Thank you in advance.