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 $$ f_d(x) = \sum_{i=0}^d \frac{(d+1+i)!}{(di)!i!(i+1)!}(x)^i. $$ 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?

3$\begingroup$ Have a look here $\endgroup$ – user111 Dec 12 '17 at 18:31

1$\begingroup$ @user111 If you want to promote your comment to an answer, then I will accept it. $\endgroup$ – Neil Strickland Dec 12 '17 at 18:40
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 $$\frac{P_{n}(x)P_{n+1}(x)}{1x},$$ 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 $x=12y$, it is straightforward to check that it gives indeed your formula on the segment $[0,1]$.