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While revising the Harmonic mean, I came across this inequality which I haven't figured out how to solve, but I think it should be the application of some known inequality. I would be very grateful if anyone could give me a suggestion how to solve it.

Consider a tuple of $n$ positive real numbers $(x_1,x_2,\dots,x_n)$ and another tuple of $n$ positive real numbers $(y_1,y_2,\dots,y_n)$. Let us denote as $\bar{x}$ the mean of the first tuple, i.e. $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$, and with $\bar{y}$ the mean of the second tuple $\bar{y} = \frac{\sum_{i=1}^n y_i}{n}$.

Is it possible to find the largest positive real number $C$ such that

\begin{equation} \frac{1}{n} \sum_{i=1}^n {\frac{1}{\frac{1}{x_i}+\frac{1}{y_i}}} \geq C \cdot {\frac{1}{\frac{1}{\bar{x}}+\frac{1}{\bar{y}}}} \end{equation}

is always verified ?

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For odd $i$ let $x_i\rightarrow0^+$ and $y_i=1$;

For even $i$ let $x_i=1$ and $y_i\rightarrow0^+$.

Thus, $C\leq0$ and the largest positive $C$, for which your inequality is true does not exist.

By the way, the reversed inequality is true for $C=1$.

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  • $\begingroup$ Thanks a lot for your answer Michael, really appreciated! $\endgroup$ – Enrico Piovano Oct 29 '18 at 12:59
  • $\begingroup$ You are welcome, Enrico! $\endgroup$ – Michael Rozenberg Oct 29 '18 at 13:00

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