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Hello. Let $C_1,C_2,C_3,q_1,q_2, \epsilon \geq 0 $ and $h \in (0,1)$ it is possible to solve this inequality for $L \in \mathbb{N}$?:
$C_1h^{q_1L}+2C_2C_3h^{q3L}\leq \epsilon$

Ive solved it for some special cases of $q_1$ and $q_2$ and im starting to wonder if its even possible to solve this straight forward.

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  • $\begingroup$ You can "unclutter" the expression as $ab^{c^L}+de^{f^L}\le1$ I guess. I doubt there is a closed-form solution. $\endgroup$ – Yves Daoust Dec 3 '18 at 17:10
  • $\begingroup$ I think so too at least i have solved it in the cases $q_1=2q_3$ and vice versa. For the other cases I will probably solve it numerically. I just thought maybe some1 has an idea that i didnt have $\endgroup$ – StefanWK Dec 3 '18 at 17:23
  • $\begingroup$ Mh, I am not sure that my retranscript is correct. Isn't it of the simpler form $ab^L+cd^L\le1$ ? $\endgroup$ – Yves Daoust Dec 3 '18 at 17:28
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The inequation can be rewritten

$$ab^L+cd^L\le1$$ with $b,d<1$.

This is a decreasing function of $L$. As $L\in N$, a simple linear search by increasing values of $L$ can do. For better efficiency, you can use an exponential search (doublings) followed by a dichotomic one.

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