Let $f:[\frac{1}{2},1]\rightarrow\mathbb{R}$ be nonnegative, monotone increasing, such that there exists $M>0$, $\alpha>0$, such that for any $\frac{1}{2}\leq r<R\leq 1$, it holds

$$f(r)\leq\frac{1}{2}f(R)+\frac{M}{(R-r)^{\alpha}}.$$

Then for some $C_{\alpha}>0$ depending only on $\alpha$, we have

$$f(\frac{1}{2})\leq C_{\alpha}M.$$