I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers $a_1 +a_2 +... +a_n = m$ and the sum of their reciprocals. In particular, I suspect that \begin{equation} \sum_{i=1}^n \frac{1}{a_i} \geq \frac{n^2}{m} \end{equation}
with equality when $a_i = \frac{m}{n} \forall i$. Are there any references or known theorems where this inequality is proven?

This interesting answer provides a different lower bound. However, I am doing some experimental evaluations where the bound is working perfectly (varying $n$ and using $10^7$ uniformly distributed random numbers).


by Cauchy schwarz inequality$$\left(\sum_{i=1}^{n}{\sqrt{a_i}}^2\right)\left(\sum_{i=1}^{n}\frac{1}{{\sqrt{a_i}}^2}\right)\ge (\sum_{i=1}^{n} 1)^2=n^2$$

WLOG $a_1\ge a_2...\ge a_n$ then $\frac{1}{a_1}\le \frac{1}{a_2}..\le \frac{1}{a_n}$

So by chebyshev's inequality $$n^2=n\left(a_1\frac{1}{a_1}+a_2\frac{1}{a_2}..+a_n\frac{1}{a_n}\right)\le \left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}\frac{1}{a_i}\right)$$

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  • $\begingroup$ @newman_ash were you able to follow the chebyshevs proof? $\endgroup$ – Albus Dumbledore Nov 22 at 10:33
  • $\begingroup$ Yes I was able to follow it $\endgroup$ – newman_ash Nov 22 at 10:41
  • $\begingroup$ But to me it is not obvious how to get rid of $\sum a_i^2$ and $\sum 1/a_i^2$ in the Cauchy-Schwarz inequality $\endgroup$ – newman_ash Nov 22 at 11:05
  • $\begingroup$ @newman_ash i edited $\endgroup$ – Albus Dumbledore Nov 22 at 11:49
  • $\begingroup$ Thank you for answering and editing. Unfortunately, I am still not getting how you are introducing the square roots. You could use the fact that $\sqrt{\sum a_i} <= \sum \sqrt{a_i}$, but you would get a less tight bound $n/m$ $\endgroup$ – newman_ash Nov 25 at 19:00

The author is using the Arithmetic Mean - Harmonic Mean ("AM-HM") inequality: https://en.wikipedia.org/wiki/Harmonic_mean#Relationship_with_other_means

This is a popular inequality in the math olympiad world; you can find a proof here: https://artofproblemsolving.com/wiki/index.php/Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality

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