Trouble with sum of inverse integers vs. inverse of sum of integers

Question

I'm having trouble trying to show the following:

$$ \frac{1}{r_{1}}+\dots+\frac{1}{r_{v}} \ge \frac{v^{2}}{r_{1}+\dots+r_{v}} $$

where $r_{i}>1$ for all i and $r_{i}$ are not necessarily equal. And: $$v=\frac{n}{r}\;;\;n=\sum_{i}r_{i}=rv$$ $$r= \frac{\sum_{i}r_{i}}{v}$$

I've run multiple simulations which all show that the required result is true. I've been able to show that the RHS has a larger denominator but also a larger numerator; which left me stuck trying to show that the effect of the denominator dominates.

Answer 1

As all $r_i \gt 1$, then they are obviously positive. Also, I assume $v$ is the number of items based on what you wrote. Thus, in the list of HM-GM-AM-QM inequalities, you can use the second & fourth ones counting from the left (i.e., the harmonic mean is less than or equal to the arithmetic mean), with their $n = v$ and $x_i = r_i$, to get after cross-multiplying & dividing,

$$\begin{equation}\begin{aligned} \frac{v}{1/r_1 + 1/r_2 + \cdots + 1/r_v} & \le \frac{r_1 + r_2 + \cdots + r_v}{v} \\ \frac{v^2}{1/r_1 + 1/r_2 + \cdots + 1/r_v} & \le r_1 + r_2 + \cdots + r_v \\ v^2 & \le \left(\frac{1}{r_1} + \frac{1}{r_2} + \cdots + \frac{1}{r_v}\right)(r_1 + r_2 + \cdots + r_v) \\ \frac{v^2}{r_1 + r_2 + \cdots + r_v} & \le \frac{1}{r_1} + \frac{1}{r_2} + \cdots + \frac{1}{r_v} \end{aligned}\end{equation}\tag{1}\label{eq1A}$$

Answer 2

It's just Cauchy-Schwarz inequality: $$\sum_{k=1}^v\frac{1}{r_v}=\sum_{k=1}^v\frac{1^2}{r_v}\geq\frac{\left(\sum\limits_{k=1}^v1\right)^2}{\sum\limits_{k=1}^vr_v}=\frac{v^2}{\sum\limits_{k=1}^vr_v}.$$