I am having difficulties to prove $\log \left[ {\sum\limits_{i = 1}^M {{\varepsilon _i}{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}} } \right]$ is convex for non-negative a, b,u. Where, $Q\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int_x^\infty {{e^{ - \frac{{{v^2}}}{2}}}dv} $.

I know the following properties of the above function:

${{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}}$ is convex.

$\sum\limits_{i = 1}^M {{\varepsilon _i}} =1$

The Range of the log function is $(0,0.5]$