Harmonic mean – Partial sum

I am an economist, and in a research paper I am working on, I came accross an harmonic mean. I am not very familiar with this kind of mean, and did my best to study the topic, but I have a few questions I am not sure how to think about (and depending on which I would be able to obtain nice economic intuitions – about how to tax incomes!).

Consider the “smooth” function $pin(0,1) rightarrow x(p)$, with $x(p)>0$, $x^prime(p)<0$ and $x^{primeprime}(p)<0$.

Let $h(p)=frac{1-p}{int_p^1 frac{1}{x(pi)} dpi}$. This is the harmonic mean of $x$ above $p$.

Because $x$ is decreasing, $h(p)$ decreases with $p$. Right?

I would like to:

  • compare the rate of increase of $x(p)$ with the rate of increase of $1/h(p)$. More precisely, is $x(p)/h(p)$ increasing or decreasing in $p$ given the above assumptions?
  • say something about whether $x (p) left(frac{1}{h(p)}-frac{1}{h(0)}right)$ is larger or lower than $1$.

Any help would be most appreciated. I am a little bit stuck right now…