Given an arbitrary number of variables $\epsilon_i$ that can be picked from a domain $[0, W]$ and some inequality relation between all the variables $G(\epsilon_1, \epsilon_2, ...)$, is there some way to use Mathematica to find the smallest possible value for $W$ such that $G$ holds true?
Example: Somewhere in the problem I'm solving I get the following relation between the values of a diagonal matrix perturbation with values $\{ \epsilon_1 ... \epsilon_4\}$ that must hold if I want certain properties of the perturbed eigenvectors to be true:
$\epsilon_1 > a(b - c \epsilon_2 - d\epsilon_3 + f\epsilon_4)$
where $a, b, c, d, f$ are positive constants. Is there a way to find the smallest bound on $\epsilon_i$, given these constants, so that I could pick values of $\epsilon_i$ that satisfy this inequality? Thanks.