In trying to solve some quantum mechanical problem, I came across a proposition which can be formulated like the following:
$$\left|\int F f + G g \; d \tau' \right| \ge \left| \int F g - G f\; d \tau' \right|$$ for
$$ \left|\int F f d \tau' \right| \gg \left| \int G g \; d \tau' \right|, $$
with everywhere real, differentiable and integrable functions $F, f, G, g$. Is the statement possibly correct and in case how to show that?
I have something like Cauchy-Schwarz in mind, but I am quite inexperienced with inequalities and hope that some here might immediately see its wrong or not.
A simple test for numbers instead of functions and integrals would not contradict the inequality: $Ff=1, Gg=-\epsilon$ yields $|1-\epsilon| \ge |\epsilon\frac{f}{g}|$.
Remark 1: Some details about which I am not sure if they are of importance in the first instance are that all these functions lets call them $\psi$ depend on the set $\tau$ of variables: $\psi(\tau)$, while integration in the inequalities takes place only in the subset $\tau'\subset\tau$. The inequalities then are meant to hold point-wise for the remaining "parameter". Also $\int F d\tau = \int G d\tau = 1$, while $g$ and $f$ are the result of acting some differential operator on $F$ and $G$.)
Remark 2: To me it appears, that in this formulation its a mathematical rather then physical problem.