TITLE: Bounds on polynomial values
QUESTION [7 upvotes]: 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
$$\vert f(x)\vert\leq \frac1{\sqrt2}(n+1), \qquad \text{for $\vert x\vert\leq1$}\,\,\,?$$

REPLY [3 votes]: Inequalities between different $L^{p}$ norms of the type $\|q_{n}\|_{p}\leq C_{n}\|q_{n}\|_{q}$, $0<q<p\leq\infty$, where $q_{n}$ is a polynomial of degree $n$ are sometimes called Nikolskii's inequalities, see p.102 of

R.A. DeVore; G.G. Lorentz,  Constructive approximation,
  Springer-Verlag, Berlin, 1993.

For the norm
$$\|f\|_{p}=\left(\int_{a}^{b}|f|^{p}dx\right)^{1/p},$$
on an interval $[a,b]$, one has
$$\|q_{n}\|_{p}\leq(2(q+1)n^{2}/(b-a))^{1/q-1/p}\|q_{n}\|_{q}.$$
The proof essentially uses Markov inequality. For $[a,b]=[-1,1]$, $q=2$ and $p=\infty$, it would give
$$\|q_{n}\|_{\infty}\leq\sqrt{3}n\|q_{n}\|_{2}.$$
About the Legendre polynomials discussed in the comments, it is known that the sum $\sum_{k=0}^{n}P_{k}^{2}(x)$, that is the reciprocal of the so-called Christoffel function, behaves, for $n$ large, like $n/(\pi\sqrt{1-x^{2}})$ for $x\in(-1,1)$, and behaves like $Cn^{2}$, $C$ some constant, at the endpoints $\pm1$.