Consider the following expression, where $H(\mathbf{x})$ is the harmonic mean of a vector $x_1, x_2, ..., x_n$.
$$\bigg(1-\frac{1}{n}\bigg)H(\mathbf{x})-\bigg(1-\frac{1}{n}\bigg)^2\frac{H(\mathbf{x})^2}{x_1}$$
Is the expression decreasing in $H(\mathbf{x})$?
It seems so: let $N$ denote $\bigg(1-\frac{1}{n}\bigg)$ and $K$ denote the harmonic mean:
$$NK-N^2\frac{K^2}{x_1}$$
$$D_k[..]= N-2N^2\frac{K}{x_1}<0$$
$$\iff 2N^2\frac{K}{x_1}>N$$
$$\iff K>\frac{x_1}{2N}$$
$$\iff H(\mathbf{x})>\frac{x_1}{2\bigg(1-\frac{1}{n}\bigg)}$$
$$\iff \frac{2(n-1)}{\frac{1}{\sum_{i=1}^n x_i}}>x_1$$
$$\iff 2(n-1)>\frac{x_1}{\sum_{i=1}^n x_i}$$
Which is true. Is my reasoning correct?