Let $X[k] \in \mathbb{C}$ be the discrete Fourier transform of $x[n] \in \mathbb{R}$, where $k,n = 0,1,2,...,N-1$.
Parseval's theorem relates the arithmetic mean (AM) of absolute value squared discrete Fourier transform points with $x[n]$ as
\begin{align*} \sum_{n=0}^{N-1}|x[n] |^2 &= \text{AM}(|X[k]|^2) \\ & = \dfrac{1}{N} \sum_{n=0}^{N-1} |X[k]|^2 \end{align*}
Does there exist a nice expression of $x[n]$ that equals the harmonic mean (HM) of absolute value squared discrete Fourier transform $\dfrac{N}{ \sum_{n=0}^{N-1} \frac{1}{|X[k]|^2}}$?
I don't think so. Consider that the harmonic mean is zero whenever $X(k)=0$ for some $k$. Thus, the conjectural "nice expression of $x[n]$" must be able to detect whether the signal lacks some frequency. In order to do that, it would have to compute the Fourier transform of $x$ in some guise.