Are posynomial functions convex?

Question

I know that you can transform a posynomial function into an exponential function, which is convex. Does this imply that all posynomial functions are convex?

Answer 1

No, that conclusion does not hold because generally, the product of two convex functions need not be convex.

As an example, $f(x, y) = x y^{-1}$ (for $x, y > 0$) is a posynomial function, but not convex: $$ f(\frac{1+5}{2}, \frac{1+3}{2}) = \frac 32 > \frac 43 = \frac 12 \left( f(1, 1) + f(5, 3)\right) $$