# upper/lower bound on ratio of square roots

I have the following function

$$f(x) = \frac{1+a\sqrt{bx-1}}{1+a\sqrt{cx-1}}$$ on $$[1, \infty)$$.

$$a > 0 \quad b, c\ge 1$$

I would like to find an upper bound on this function, one that is tight when $$b = c$$.

I have tried various transformations, looking at the derivative, but to no avail.

My hope was that the extrema are at $$x = 1$$ and $$x = \infty$$, but numerical simulations contradict this.

Any help would be appreciated.

• Yeah, $f(x)$ has either a minima or a maxima over $(0,\infty)$, depending on the relative magnitude of $b$ and $c$ . It'd take a lot of work, though. – Quanto Oct 31 '19 at 17:36