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Let $a,b$ and $c$ three positive reals numbers such that $abc=1$. Define the function $f$ by $f(x)=\frac{^1}{1+(n-1)x^n}$ where $n$ is a positive integer. Prove that

$$a^2f\left(\frac{b}{a}\right)+b^2f\left(\frac{c}{b}\right)+c^2f\left(\frac{a}{c}\right)\ge\frac{3}{n}$$

I tried Jensen's inequality but got nothing.

Thanks for any help.

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  • $\begingroup$ According to your assumption, this is not true e.g. take $ a=b =c=0.1$ then we have $$3a^2f (1)=\frac{3a^2}{n} \not\ge \frac{3}{n} $$ $\endgroup$ Jul 25, 2014 at 21:52
  • $\begingroup$ Your example doesn't fulfill the condition $abc=1$. $\endgroup$
    – Abou Salah
    Jul 26, 2014 at 4:25
  • $\begingroup$ Right, I didn't take care from this condition. Sorry $\endgroup$ Jul 26, 2014 at 6:33

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