Let $p_1>p_2$ and $n_1>n_2$ be positive numbers. I want to show that,
$ \frac{\log \left(\frac{p_1}{n_1}+1\right)}{\log \left(\frac{p_2}{n_2}+1\right)}\leq \frac{\log \left(\frac{p_1}{c+n_1}+1\right)}{\log \left(\frac{p_2}{c+n_2}+1\right)} $
where $c$ is a positive number. The simplest solution I can imagine of is to define,
$ f(u)=\frac{\log \left(\frac{p_1}{n_1+u}+1\right)}{\log \left(\frac{p_2}{n_2+u}+1\right)} $
and show $f(u)$ is an increasing function over $u \ge 0$. We have,
$ \frac{\partial{f}}{\partial{u}}=\frac{\frac{p_2 \log \left(\frac{p_1}{n_1+u}+1\right)}{\left(n_2+u\right) \left(n_2+p_2+u\right)}-\frac{p_1 \log \left(\frac{p_2}{n_2+u}+1\right)}{\left(n_1+u\right) \left(n_1+p_1+u\right)}}{\log ^2\left(\frac{p_2}{n_2+u}+1\right)} $
Since, the denominator is obviously positive, we need to show the numerator is also positive. But I cannot do that. Any help on the issue is appreciated. Any other (simpler) way to prove the first inequality is also welcomed.
EDIT: In addition to the above assumptions, we have $u \le p_2$. So we need to show the inequality over $0 < u \le p_2$.