Product of convex and concave functions

Question

I'm struggling with a problem which has been asked a lot of times but always in different versions so I'm still not quote sure about my answer. I have a function l:R→R:x↦l(x) which is the result of different smaller functions namely: $$l(x)=f(x)g(x)-g(x)h(x)$$ and $0<x<1$, with $f(x)>0$;$g(x)>0$ and $h(x)>0$.

Let $f(x)$ and $g(x)$ be convex as well as decreasing over the range of $x$ and let $h(x)$ be concave and also decreasing. Is the following statement true?

1. The product of $f(x)$ and $g(x)$ is concave and the product of $g(x)$ and $h(x)$ is concave.

2. But because of $-g(x)h(x)$ the second term of $l(x)$ is convex

3. We are only interested in positive values for $l(x)$ we can say that $f(x)g(x)>g(x)h(x)$, $l(x)$ is concave for all $l(x)>0$

Thank you.

Answer 1

The claim is false in general.

However, Exercise 3.32.(a) on p. 119 in the book quoted by Convexity of the product of two functions in higher dimensions provides the following sufficient condition to obtain your result: $f,g: \mathbf{R} \rightarrow \mathbf{R}$ are convex, both nondecreasing (or non increasing) and positive on an interval.

A condition (stronger than convexity) that directly addresses your issue is log-convexity. The set of log-convex functions is closed under product, sum and positive scaling.

Answer 2

Thank you for your answer. I was checking my formula again for different parameter values and it looks pretty bad for my original claim that $l(x)$ is concave, which is off the table now. In general $l(x)$ can look like this (again for different parameter values): https://ibb.co/eKY16F

The original problem (maybe I should make another thread for this) is the following: There is a value for $x$ that maximizes and minimizes $l(x)$ in the first quadrant (so for non-negative values). The range for $x$ is a compact set with $x[0,x°]$ where $x°$ is the zero of the function $l(x)$ which again depends on different parameter values. Using the theorem of Weierstrass I can say that: (I) That there exists a maximum and a minimum for $l(x)$ with the relevant $x$ values in the compact set $x[0, x°]$.

The Problem is: I can't really say something about the uniqueness of the maximum or the minimum of $l(x)$. I can't use the Karush-Kuhn-Tucker-Theorem because $l(x)$ is not concave. Maybe I should check for quasi-concavity?