How to check whether a complex multivariable function is convex? 
Given a convex function $f : \mathbb R^2 \to \mathbb R$ and real numbers $a$ and $b$, I want to check whether the following function $g : \mathbb R^2 \to \mathbb R$ is convex.


First, I think I should use the lemma below:



So I start to write down g(αx1 + (1-α)y1 , αx2+(1-α)y2)
what I get is like this:

Then I get stuck, the function I get is just too complex and I don't know how to do next. Did I do it in the wrong way? Can anyone give me some hints?

Hints:


*

*Sum of convex functions are convex.

*Non-negative multiple of a convex function is convex.

*Composition of a convex function and an affine function is convex.

*Maximum of convex functions is convex.

