I have the following inequality: $n \ge \frac{K_n^2}{\epsilon^2} \frac{\log K_n}{\epsilon},\text{ where }K_n = (\log n)^3.$
I would like to solve it, even numerically.
I thought that numerically it can be solved by iteratively setting $K_n$ using a current value for $n$ and then $n$ using a current value for $K_n$.
However, I get a weird behavior. For different $\epsilon$ (more specifically for $\epsilon = 3.3409202$) I get that the answer for $n$ is 2.0127e+06. If I increase $\epsilon$ by a bit, I get a complex number. For other similar inequalities, I would get a different, but similar interesting "boundary" behavior: there would be a certain epsilon under which the value of solved $n$ will be very large, and over which the $n$ will suddenly jump to a really small value.
Is there any explanation for what's going on here? Is there a better way to solve this inequality?
Thanks.