Given $b > 0$, let $f : \Bbb R_{>0}^2 \to \Bbb R$ be
$$f(x,y) := \frac{x}{y \log_2 \left(1 + b \frac{x}{y}\right)}$$
I would like to prove that $f$ is convex.
I tried to use the definition of convexity (i.e., the Hessian matrix) but it seems to be non-convex. Is there any transformation I could do to make my function convex?
$f(x,y)$ is not convex. For $b=1$ try points $A=[10,1]$ and $B=[30,1]$ and the value at the midpoint:
$f(A) = 2.8906482,~f(B) = 6.0554725,~f\left(\frac{A+B}{2}\right) = 4.5534049 > \frac{f(A)+f(B)}{2} = 4.4730604$