Q:

Strict convex function?

I try to prove that 
$g(x)= K |x|^2/2 + z(x)$ is strictly convex, given that $z(x) \geq - m(1 + |x|^p)$ with $m \geq 0$, $0 \leq p \leq 2$, forall $x \in \mathbb{R}^n$, provided $K$ is sufficiently large.
How to show that? 

A:

The conclusion is false. For example, let $n=1$ and $z(x)=x^2(1+\sin x)\ge 0$. Then no matter how large $K$ is, $f''((2k+\frac{1}{2})\pi)<0$ when $k\in\mathbb{N}$ is large.

