I've 2 functions:
$$f_1(p, h) = \frac{1}{p - p(1-h)^{17+16c_0})},$$ and $$f_2(p, h) = \frac{1}{ph}.$$ Further, I've the following constraints on variables: $ 0 < p \leq 1, 0 < h < 1$ and $c_0$ is a constant with $c_0 > 1$.
I've the intuition that $f_1 > f_2$ under these settings. However, I'm not sure how to prove it. Also, if my intuition is incorrect, how can I at least find the intervals in which $f_2 > f_1$ ?
Note that $\frac {f_2}{f_1}=\frac {1-(1-h)^{17+16c_0}}h$ so the value of $p$ does not matter. We can then write $(1-h)^{17+16c_0} \gt 1-h(17+16c_0)$ by the binomial theorem so as long as $17+16c_0 \gt 1$ you will have $f_2 \gt f_1$