I have some type of proof for the inverse inequality: $|\nabla v |_{H^1} \le C |v|_{H^1}$
This proof uses the following property for quasi-uniform triangulations: $ \frac{\int_{{K}^\wedge}{|\nabla v^\wedge |}^2}{\int_{K^\wedge}|{v}^\wedge |^2} \le C$
the wedge refers to the reference triangle. So $K^\wedge$ is the reference triangle and $v^\wedge$ is the test function on the reference triangle. I'm not sure what type of semi-norm it is, but I assume it's either $L^2$ or $H^1$.
I do not understand why this property is true. Any help appreciated.