Estimating Parameters of Log-Concave Model

Question

I'm considering the problem of fitting some given data,

$(x_1,y_1), (x_2, y_2), \dots (x_m,y_m)$,

with a log-concave function,

$y = (\alpha x)^\beta$.

Parameters that I want to estimate are $\alpha$ and $\beta$. I tried least-squares method on the log of data:

$min. \|\log(y) - \beta(\log(\alpha) + \log(x))\|_2^2$,

but this doesn't seem to be a convex problem.

Any suggestions?