Sum over Product Lemma



I was doing some math with Lars and during our session, we came up with the following inequality. Let $a_1,\ldots,a_r\in\mathbb N$ be integer numbers with $a_i\ge k$. Then, \[ \frac{\sum_{i=1}^r a_i}{\prod_{i=1}^r a_i} = \sum_{i=1}^r \frac{1}{\prod_{j\ne i} a_j} \le \frac{r}{k^{r-1}}= \frac{rk}{k^r} \] The inequality is completely obvious, but I found it useful and didn't want to forget about it.

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