An inequality for a two variable function

[amirmath]

Suppose that $F(u,v)=a|u+v|^{r+1}+2b|uv|^{\frac{r+1}{2}}$, where $a>1, b>0$ and $r\geq3.$ How we can show that there exists a positive constant c such that
$F(u,v)\geq c\Big( |u|^{r+1}+|v|^{r+1}\Big).$

 

[Simon Bridge]

You'd start by considering what sort of curve is described by the part in parentheses in the second relation.
What role does c play? Is there a minimum value that the first expression can take?