Any suggestions on proving the convexity of
$f (\boldsymbol{x}) = \log \left ( 1 + \left ( \sum_i \frac {1}{\sqrt{x_i}} \right )^2 \right )$
where $\boldsymbol{x} = \left [x_1~x_2~\dots~x_L \right ]$ and $x_i > 0, \forall i$?
A straightforward way is to calculate the Hessian and check the semidefiniteness but that becomes ugly fast. Is there a simpler way?
UPDATE: We can express $f (\boldsymbol{x})$ as $f (\boldsymbol{x})=g(h(\boldsymbol{x}))$ where $$g(z)=\log(1+z)$$ and $$h(\boldsymbol{x})=\left ( \sum_i \frac {1}{\sqrt{x_i}} \right )^2.$$ We know that $g(z)$ is quasiconvex and that $h(\boldsymbol{x})$ is convex. Thus, the result to the original question would follow if the following is true:
Is a quasiconvex function of a convex function convex?