Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Here's a homework question I'm struggling with:

Prove/disprove the next statement:

Let $f,g$ two convex functions, then $h(x)=f(x) \cdot g(x)$ is also convex

So, we know that $h'(x)=f'(x) \cdot g(x) + f(x) \cdot g'(x)$. We also know that $f'(x),g'(x)$ are monotonically increasing because they are convex. If I can show that $h'(x)$ is also monotonically increasin I'm done, but I'm not sure how to do it. Any hints?


share|cite|improve this question
Try to find a counterexample ... :D –  martini May 21 '12 at 8:14
That didn't help me too mcuh, sorry –  yotamoo May 21 '12 at 8:25
Ok ... $x \mapsto (x-a)^2$ is convex for any $a$, right :) –  martini May 21 '12 at 8:28

2 Answers 2

up vote 4 down vote accepted

Hint: You can write the function defined by $x\mapsto-x^2$ as the product of two very simple linear and hence convex functions.

share|cite|improve this answer
So the statement is not even true. I assumed it is. Thanks! –  yotamoo May 21 '12 at 8:37
I am sorry but I am new to convex functions. Is the statement "the product of convex functions is convex" true or false ? Could you please clarify and give more details. For example if $f(x)=-x^2$ and $g(x)=-x^2$ then $fg$ is a convex function whereas $f$ and $g$ are not. –  roni Aug 25 '14 at 7:41
@roni It is false. Take the two functions $x\mapsto x$ and $x\mapsto -x$. –  Michael Greinecker Aug 25 '14 at 16:40

The functions $f(x)=1-x$ and $g(x)=1+x$ are convex. However, their product $(f*g)(x)=1-x^2$ is not. So it hoes not hold.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.