Let $x_1, \dots, x_N \in (0, \infty)$ each be a D5407: Positive real number.
A D993: Real number $a \in (0, \infty)$ is a harmonic mean of $x_1, \dots, x_N$ if and only if \begin{equation} \sum_{n = 1}^N a^{-1} = \sum_{n = 1}^N x^{-1}_n \end{equation}