I am trying to show whether this function is convex (for $f:\mathbb{R}_{++}^n \to \mathbb{R}$): $$ f = \log(x_1^\alpha \cdot x_2^\alpha \cdot ... \cdot x_n^\alpha),$$ where $\alpha > 0$
I've tried to show whether $f$ is convex via the definition of convex functions, but I've had no luck thus far. Is there a better way to do this? Has anyone been successful showing whether this function is convex?
Thank you!
With $x_2, \ldots, x_n$ held fixed, the function is of the form $x_1 \mapsto c + \alpha \log x_1$, so $f$ is not even component-wise convex.