Let $g$ be a quadratic form of a bilinear function that is given by $g(x,y) = -x^2+y^2$. Suppose I have another quadratic form of another bilinear function given by $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ for which $f \geq 0$ whenever $|x|=|y|$.
- how do I demonstrate the bound $f \geq c |g|$?
- how do I generalize it from two-dimensional space to general dimensions?
My attempts:
- Since $f$ and $g$ are both quadratic forms of bilinear functions, I have tried to use their matrix representation. I had two options: choose a basis such that $f$ is (at least for the case when it is symmetric and therefore) diagonal, or choose a basis such that $g$ is diagonal. Next, I considered two cases: $g(x,y) > 0$ or vice versa. Then I considered the difference between two matrices $x^t [f] x - x^t [g] x$ and try to test for positive definiteness equivalent to eigenvalue nonnegativity.
- I have try to bound the ratio $$\dfrac{f(x,y)}{|g(x,y)|}$$ and derive $c$ as infimum.
None of my attempts lead me anywhere.