Formalizing (dollar) cost averaging

Question

Let $P$ be a strictly positive real-valued function on $[0,1]$ (with however many niceties you want to impose on it). Is there anyway we can quantify that the following inequality is usually true,and that when it isn't true it usually isn't off by much? $$\int \frac{1}P>\frac{1}{P(t)}$$ If $P$ represents the price of an asset and you have ${$}1$, then the left side should represent the amount of the asset you get from dollar cost averaging over infinitesimal intervals (i.e the limit of DCAing over smaller and smaller intervals) while the right side represents the amount of asset obtained buying at a fixed time $t$.