This is from the book The Cauchy Schwarz masterclass. Could someone explain why the inequality mentioned holds in the passage below to me? Note that he meant quadratic formula instead of binomial foruma (typo) and $S\subset \mathbb{R}^2$. Since $(2B)^2>4AC$ implies the polynomial has real roots. Does this mean there are no real roots?
Embarrassedly, I can't believe I didn't get this earlier. Admittedly, my high school maths are very rusty.
Anyway, so we know $$f^2(x,y)\geq 0 \implies A=\int\int_S f^2(x,y) dxdy\geq 0$$ Thus, this is a "U"-shape parabola and has one global minimum. We are given $$p(t)\geq 0$$ so the curve $p(t)$ is above the $t$-axis. So the discriminant is non-positive. Thus, $$(2B)^2\leq 4AC\implies B^2\leq AC.$$
Finally, $$B^2=AC\iff \Big(\int\int_S f(x,y)g(x,y)dxdy\Big)=\int\int_S f(x,y)^2dxdy\int\int_S g(x,y)^2dxdy$$ which is only true if $f(x,y)=\lambda g(x,y)$.