# What is the most general distribution for which E[1/x] = 1/E[x]?

What is the most general distribution for which $E\left[1/x\right]=1/E\left[x\right]$?
What is the most general distribution for which the expected value of the multiplicative inverse equals the multiplicative inverse of the expected value?
Motivation: I'm into modelling dynamics on graphs and I found a problem which is easily solvable in cases where the degree distribution of the vertices is a distribution where $E\left[1/k\right]=1/E\left[k\right]$. (${k}_{i}$ is the degree of the ith vertex) From this solution I may gain an insight into how to unify multiple models.
So particularly I'm looking for a distribution which consists of non-negative, finite integers. But I'm also interested in continuous solutions. Distributions where $E\left[1/{k}^{n}\right]=1/E\left[{k}^{n}\right]$ may also help unifying the models.
What I do know so far, that ${k}_{i}=1$ is a particular solution. In the continuous case every function where $f\left(x\right)=f\left(1/x\right)$ and $E\left[x\right]=1$ is a solution. I know what momentum generating functions are and they seem like a good direction to try in, but I failed so far.
What is the most general form of this distribution? Does it have a name? It sounds like something trivial, like a "famous" distribution, but I can't find it.
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Explanation:
By Jensen's inequality applied to the convex function $f\left(x\right)={x}^{-1}$ on $\left(0,\mathrm{\infty }\right)$,
$\frac{1}{\mathbb{E}\left[X\right]}<\mathbb{E}\left[\frac{1}{X}\right]$
for any non-constant nonnegative random variable X. Thus, the constant random variable is the only such random variable.