Suppose we have two functions R(x) and T(x). Assume the functions to be smooth (infinitely differentiable; if you can give an answer without the differentiability condition would be great as well) consider the inequality:
There exist some values of r0 for which the inequality holds true for all values of x. I assume these values form an interval and denote rmax to be the greatest value in that interval. i.e. the largest value of r0 for which the function remains bound by -0.5 and 0.5. I want to express rmax as a functional of R(x) and T(x).
I tried to make a coordinate transformation
so that all functions that do not satisfy the inequality end up being undefined for some x and then checking the radius of convergence to find rmax, but the derivatives of the function in the new coordinates ended up so messy. Any help is appreciated