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Part of proof

  1. Sep 11, 2010 #1
    Suppose that [tex]f[/tex] is a real-valued continuous function on [tex][0,1][/tex] and let [tex]F_n=\int\limits_{0}^1 f^n(x)\ dx[/tex].

    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] ??
  2. jcsd
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