For $p\in (0,1)$ and $\epsilon>0$ a small enough constant, consider the function $f:\mathbb{N}\to\mathbb{R}$ given by

$$f(y)=\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt.$$

Is it true that $f(y)=O(1)$ for all $p\in(0,1)$? Numerical experiments seem to support this assertion.

A possible first approach would be to compute the Taylor expansion of the numerator around $t=0$. We have $$ \left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}=O_y(t^2),$$ where $O_y$ hides terms which depend on $y$. If these terms were independent of $y$ then the result would follow easily, but this is not the case.