# Part of proof

Suppose that $$f$$ is a real-valued continuous function on $$[0,1]$$ and let $$F_n=\int\limits_{0}^1 f^n(x)\ dx$$.

If $$F_3=0$$, then by Cauchy-Schwartz inequality $$\left(\int_{0}^{1}(a+f^{2}(x))f(x)dx\right)^{2}\leq\int_{0}^{1}(a+f^{2}(x))^{2}f^{2}(x)dx, a\in \mathbb R$$.

This implies $$\ast$$.

$$-a^{2}(1-\frac{F_{1}^{2}}{F_{2}})-2aF_{2}\leq F_{4}\cdots\cdots\cdots\ast$$

I don't understand how we get $$\ast$$ .

The calculations gives me $$a^{2}F_{1}^{2}-a^{2}F_{2}-2aF_{4}\le F_{6}$$ ??