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Let $f:\mathbb{R}^n\to\mathbb{R}$ be a real-valued, $n$-dimensional function defined as follows:

$$ f(\mathbf{x}) = g(\mathbf{x})h(\mathbf{x}), $$

where $g:\mathbb{R}^n\to\mathbb{R}$ is convex, and $h:\mathbb{R}^n\to\mathbb{R}$ is bounded, for instance $h(\mathbf{x})\in[a,b]$, $a,b\in\mathbb{R}$, $0<a<b$, $\forall\mathbf{x}\in\mathbb{R}^n$.

What is true about the convexity of $f$? What if $g$ is quasi-convex?

Any helpful comment would be nice! Thanks!

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Nothing can be said in general without further conditions on $h$. For instance, if $h(\mathbf{x})=-1$ then $f$ is concave. Even in $n=1$ nothing can be said. Consider for example $g(x)=x^2$ and $h(x)=x$.

Assuming $h$ positive, $0< h(x)< 1$ for instance, does not help. Take for instance $g(x)=1$ and $h(x)=(1+a\sin x)/2$ with $0<a<1$.

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Yeah, let's suppose that $h$ is positive, for instance, $0<h(\mathbf{x})<1$, $\forall \mathbf{x}\in\mathbb{R}^n$. What then? – nullgeppetto Jan 17 '14 at 15:13

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