For Schwarz inequality for 2 signals $s_1(t)$ and $s_2(t)$, equality holds if $s_1(t) = c\cdot s_2(t)$ ($c$ is constant). Does $c$ have to be real?
Schwarz Inequality for continuous-time Complex valued functions is given as follows:
$$\left|\int^{\infty}_{-\infty}f(t)^* \cdot g(t) dt \right|^2 \le \int^{\infty}_{-\infty}\left|f(t)\right|^2dt \cdot \int^{\infty}_{-\infty}\left|g(t)\right|^2dt$$
As you can see that, on left hand side magnitude is taken after integration making it real number and the quantity on right hand side is also real number because magnitude-squared is taken before integrating. Thus, the comparison is valid and can be done for all Complex-valued continuous time functions which are Square integrable. Meaning the Inequality  exists for all functions $f(t)$ satisfying the following:
$$\int^{\infty}_{-\infty}\left|f(t)\right|^2dt = c &lt; \infty$$