Prove that $x\log{(-x/f(y))}$ is convex

Question

This is exercise B.16 from Fundamentals of Convex Analysis by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal.

Let $f: (a, b) \to (-\infty, 0)$ to be convex, prove that $$h: (0, \infty)\times(a, b), h(x,y):= x\log{\frac{-x}{f(y)}}$$ is convex.

Answer 1

It is straightforward to show that $x\log(-x/y)$ is convex, and nondecreasing in $y$ ($y<0$). Therefore, substituting the convex function $f$ for $y$ does not affect convexity (Boyd&Vandenberghe, Section 3.2.4).