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If [tex]F_3=0[/tex], then by Cauchy-Schwartz inequality [tex]\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[/tex].

This implies [tex]\ast[/tex].

[tex] -a^{2}(1-\frac{F_{1}^{2}}{F_{2}})-2aF_{2}\leq F_{4}\cdots\cdots\cdots\ast[/tex]

I don't understand how we get [tex]\ast[/tex] .

The calculations gives me [tex]a^{2}F_{1}^{2}-a^{2}F_{2}-2aF_{4}\le F_{6}[/tex] ??