Let the polynomial $p(x)= a_0 + a_1 x + . . . + a_n x^n$ have coefficients satisfying the relation $$\sum_{i=1}^{n} a_i^{2} = 1$$
Prove that $\int_{0}^{1} |p(x)| dx \leq \frac{\pi}{2} $.
I don't have any idea to prove this inequality, is there any reference to study about integrating polynomial ?
For $x\in [0,1)$, by Cauchy-Schwarz inequality, $$|p(x)|^2\leq \sum_{i=0}^{n} a_i^{2}\cdot \sum_{i=0}^{n} x^{2i}\leq \frac{1}{1-x^2}.$$ Hence $$\int_{0}^{1} |p(x)| dx \leq \int_{0}^{1} \frac{1}{\sqrt{1-x^2}} dx=\frac{\pi}{2}.$$
P.S. I am assuming that $\sum_{i=0}^{n} a_i^{2} \leq 1$. If we have the weaker condition $\sum_{i=1}^{n} a_i^{2} = 1$ the inequality does not hold. Take for example $p(x)=2$.