I am reading a Luo-Sarnak's paper "Mass Equidistribution for Hecke Eigenforms". There is this bound that I think should be easily seen, but I just don't see it. The bound is between equations (4.5) and (4.6) The paper says
We see that if $c\gg K^{\theta+\epsilon}$, then $$ \frac{4\pi\sqrt{r_1r_2(r_1+\frac{m}{d})(r_2+\frac{m}{d})}}{c}(K^{1-\theta})^{-4}\ll K^{-3\epsilon} $$
For the above, I should remark that $\theta<\frac{2}{3}$ and $\epsilon$ has been choosen such that $\epsilon<\frac{2}{3}-\theta$. The numerator of the fraction I don't think is important for this bound (as the fraction is going to be an input into a Bessel function, and we have $m,d$ are fixed integers and $r_1,r_2$ are being summed over all integers, so they won't affect $K$, but if this bound is not clear, then maybe there is a possibility that the numerator might be affecting things in a way I don't understand).
Now what I tried is just ignoring the numerator, and applying the bounds we know $$ \frac{(K^{1-\theta})^{-4}}{c}\ll \frac{K^{4\theta-4}}{K^{\theta+\epsilon}}=K^{3\theta-4+\epsilon}\ll K^{3(\frac{2}{3}-\epsilon)-4-\epsilon}=K^{-2-4\epsilon} $$ where the first $\ll$ comes from applying $c\gg K^{\theta+\epsilon}$, then the equality is just putting everything in the numerator. Finally, the second $\ll$ comes from applying the inequality that $\theta<\frac{2}{3}-\epsilon$, then we just collect terms to get to the end.