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A logarithmic inequality

Let x_1, x_2, \dots, x_n be n \geq 2 positive numbers other than 1 such that x_1^2+x_2^2+\cdots +x_n^2=n^3. Prove that

    \[\frac{\log_{x_1}^4 x_2}{x_1+x_2}+ \frac{\log_{x_2}^4 x_3}{x_2+x_3}+ \cdots + \frac{\log_{x_n}^4 x_1}{x_n+x_1} \geq \frac{1}{2}\]

Solution

The Engels form of the Cauchy – Schwartz inequality gives us:

    \begin{align*} \sum \frac{\log_{x_1}^4 x_2}{x_1+x_2} & \geq \frac{\left (\sum \log_{x_1}^2 x_2 \right )^2}{\sum (x_1+x_2)} \\ &= \frac{\left ( \sum \log_{x_1}^2 x_2 \right )^2}{2\sum x_1} \\ &\!\!\!\!\!\!\overset{\text{AM-GM}}{\geq } \frac{\left [ n \left (\prod \log_{x_1} x_2 \right )^{2/n} \right ]^2}{2\sum x_1} \\ &\!\!\!\!\overset{\text{C-B-S}}{\geq } \frac{n^2}{2n^2} \\ &= \frac{1}{2} \end{align*}

and the inequality is proven.

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