Is $f(x,y)=y\left(2^{\frac{x}{y}}-1\right)$ a strictly convex function when $x\ge0,y\ge0$?

Question

Is function $$f(x,y) = y\left(2^{\frac{x}{y}}-1\right)$$ strictly convex when $x\ge0,y\ge0$?

I can show its Hessian matrix is positive semidefinite, but it is only a sufficient condition for the strictly convex. Any help is appreciated.

Answer 1

Hint: Consider the line $y=cx$ for constant $c$.