I'm studying the following function, defined for $ x > 0 $, where all constants $ c_1, c_2, c_3, c_4, c_5 $ are positive real numbers: $$ f(x) = \frac{c_1}{x} + \frac{c_2}{\log_2\left(1 + \frac{c_3}{c_4 x + c_5} \right)} $$
I would like to determine:
Is $ f(x) $ convex on $ (0, \infty) $? Does it have a global minimum over this domain?
What I've tried:
- I know that $\frac{c_1}{x}$ is convex for $x > 0$ and diverges to infinity as $x \to 0^+$, which suggests the function is coercive.
- The second term is more complicated: it involves the reciprocal of a base-2 logarithm applied to a rational function.
- I suspect that the logarithmic expression inside is increasing and concave (as functions of the form $\log(1 + \frac{a}{x})$ often are), but taking the reciprocal complicates the analysis of convexity.
- I attempted to differentiate the function to analyze the second derivative, but the expressions became quite messy, and I wasn't confident how to proceed further to prove convexity.
Is there a more elegant or tractable method (e.g., using convexity composition rules or known properties) to determine whether this function is convex? And if convexity holds, does coercivity imply the existence of a unique global minimum over $(0, \infty)$?
Any suggestions or insights would be appreciated.