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?



Reference request: maximal ratio of different norms of polynomials
So, I write this as an answer rather than a comment to close the question.


This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is 
$$\frac{P\_{n}(x)-P\_{n+1}(x)}{1-x},$$
where $P\_{n}$ is the Legendre polynomials of degree $n$ (with normalization $\int P\_{n}^{2}(x)dx=2/(2n+1)$).
Making use of the explicit formula
$$P\_{n}(x)= \sum \_ { k = 0} ^ { n } \binom{ n } { k }\binom{ n + k }{ k } \left( \frac { x - 1} { 2} \right) ^ { k },$$
and the change of variables $x=1-2y$, it is straightforward to check that it gives indeed your formula on the segment $[0,1]$.


