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I have been studying inequalities, and I have reached Panaitopol's Inequality (Application of Cauchy-Buniakovski inequality).

$$\frac{x_1^p}{y_1^{p-1}}+\frac{x_2^p}{y_2^{p-1}}+...+\frac{x_n^p}{y_n^{p-1}} \geq \frac{(x_1+x_2+...+x_n)^p}{(y_1+y_2+...+y_n)^{p-1}}$$

It looks somewhat similar to Titu's lemma. I have noticed that if we let $y_1=1$, then we get a particular case of the Power mean inequality, $$\sum_{i=1}^n x_i^p \geq n^{1-p} \bigg(\sum_{i=1}^n x_i\bigg)^p$$

How can I prove this?

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For positive variables and $p\geq1$ by Holder we have: $$\sum_{i=1}^n\frac{x_i^p}{y_i^{p-1}}\left(\sum_{i=1}^ny_i\right)^{p-1}\geq\left(\sum_{i=1}^nx_i\right)^p$$ About Holder see here: https://math.stackexchange.com/tags/holder-inequality/info

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    $\begingroup$ Thanks very much! $\endgroup$ Commented Aug 18, 2021 at 23:59

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