How prove this inequality $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{3}}+\cdots+\frac{x_{n-1}}{x_{n}}+\frac{x_{n}}{x_{1}}-n\le \cdots$

Question

let $x_{i}\in R^{+}$, and such $$x_{1}+x_{2}+\cdots+x_{n}=n$$

show that $$\dfrac{x_{1}}{x_{2}}+\dfrac{x_{2}}{x_{3}}+\cdots+\dfrac{x_{n-1}}{x_{n}}+\dfrac{x_{n}}{x_{1}}-n\le\left(\dfrac{n}{n-1}\right)^n\left(\dfrac{1}{x_{1}x_{2}\cdots x_{n}}-1\right)$$

I think this inequality can use the EV(equal variable) method :see http://www.docin.com/p-683093738.html

But I failed.Thank you

See this. The inequality failed at 3. – sato Dec 24 '20 at 02:42 — sato, Dec 24 '20 at 02:42

score 1 · Answer 1 · answered Aug 07 '17 at 15:56

1

No, you can not use EV method here because your inequality is not symmetric and you can't make it symmetric.

answered Aug 07 '17 at 15:56

Michael Rozenberg

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How prove this inequality $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{3}}+\cdots+\frac{x_{n-1}}{x_{n}}+\frac{x_{n}}{x_{1}}-n\le \cdots$

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