# Inequality between two sums

As part of a research problem I am working on, I need to show the following inequality. Let $$x=(x_1,\dots,x_K)$$ with $$x_i > 0$$ for all $$i$$. Then, I wish to show that $$\frac{1}{K^2}\sum_{i=1}^K x_i \geq K \left(\sum_{i=1}^K x_i^{-1/2}\right)^{-2}.$$ For $$K=2$$ this is easy to show with simple algebra, but for the general case I haven't managed to find a proof. Note that this inequality can be rewritten as $$\sum_{i,j,k}^K \frac{x_k}{\sqrt{x_ix_j}} \geq K^3.$$ Numerically it always seems to clearly hold. Using the fact that the $$1/2$$-norm is higher than the $$1$$-norm, I could show that it is higher than $$K^2$$, but not any $$K^3$$.

Any help would be greatly appreciated!

Your sum $$S$$ also equals $$\sum_{i,j,k}\frac{x_j}{\sqrt{x_ix_k}}\qquad\text{and}\qquad\sum_{i,j,k}\frac{x_i}{\sqrt{x_jx_k}}.$$ Therefore $$3S=\sum_{i,j,k}\left(\frac{x_i}{\sqrt{x_jx_k}}+\frac{x_j}{\sqrt{x_ix_k}}+\frac{x_k}{\sqrt{x_ix_j}}\right)\ge 3K^2$$ on applying AM/GM to each summand.
The inequality is equivalent to $$\left(\frac{1}{K}\sum_{i=1}^K x_i^{-1/2}\right)^{-2} \le \frac{1}{K}\sum_{i=1}^K x_i$$ and that is a special case of the generalized mean inequality: $$M_p(x_1, \ldots, x_K) \le M_q (x_1, \ldots, x_K)$$ for $$p = -\frac 12 < q = 1$$.
Because by Holder: $$\left(\sum_{i=1}^K\frac{1}{\sqrt{x_i}}\right)^2\sum_{i=1}^Kx_i\geq\left(\sum_{i=1}^K\sqrt{\left(\frac{1}{\sqrt{x_i}}\right)^2x_i}\right)^3=K^3.$$