Tighter inequality than Cauchy - Schwarz inequality
I think you mean $\mathcal{E}\geq0$, otherwise $a=b$ will give a counterexample.
For example: $$2(a^2+b^2)-\frac{(a-b)^4}{a^2+b^2}\geq(a+b)^2.$$
For three variables we have the following stronger than C-S inequality.
Let $a$, $b$ and $c$ be positive numbers. Prove that: $$3(a^2+b^2+c^2)-(a+b+c)^2\geq\frac{25(a-b)^2(a-c)^2(b-c)^2}{a^4+b^4+c^4}.$$
If we'll change $25$ on $26$ we'll get a wrong inequality.