# 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…

Thanks!