Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} Put $$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i. $$ Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$. These facts must surely be known. Can anyone point me to a reference? Do the polynomials $f_d(x)$ have a standard name?

Let us consider polynomials as functions on $[0,1]$, and so define \begin{align*} \|f\|_2 &= \sqrt{\int_0^1f(x)^2\,dx} \\ \|f\|_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}. \end{align*} I am interested in the ratio of these norms. It is easy to see that $\|f\|_2\leq\|f\|_\infty$, with equality only for constant polynomials. In the opposite direction, put $$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(d-i)!i!(i+1)!}(-x)^i. $$ Experiments make it clear that $\|f_d\|_2=1$ and $\|f_d\|_\infty=(d+1)$ and that $f_d$ maximises the ratio $\|f\|_\infty/\|f\|_2$ among polynomials of degree $d$. These facts must surely be known. Can anyone point me to a reference? Do the polynomials $f_d(x)$ have a standard name?