Assume $f(x)\in\Bbb{R}[x]$ is a polynomial of degree $n$. 


**Question.** If $\int\_{-1}^1f^2(x)\,dx=1$, is it true that
$$\vert f(x)\vert\leq \frac1{\sqrt2}(n+1), \qquad \text{for $\vert x\vert\leq1$}\,\,\,?$$



Bounds on polynomial values
This is problem VI.103 in volume 2 of Polya and Szego, where they also characterize the extremal polynomials.


