Simple but difficult to prove inequality involving integrals

Solution 1:

Your inequality says $\left| \int_0^\pi e^{i\phi(x)}\ dx \right| > 2$. In fact if $|\phi'| < 1$ we have $|\phi(x) - \phi(\pi/2)| < |x - \pi/2|$ for $0 < x < \pi$, so $$\left| \int_0^\pi e^{i\phi(x)}\ dx \right| \ge \text{Re}\ e^{-i \phi(\pi/2)} \int_0^\pi e^{i \phi(x)}\ dx = \int_0^\pi \cos(\phi(x)-\phi(\pi/2))\ dx > \int_0^\pi \cos(x - \pi/2)\ dx = 2$$ Here I'm using the facts that $\cos$ is even and is decreasing on $[0,\pi]$. No smoothness of $\phi$ is necessary, just differentiability.

More generally, if you replaced $|\phi'| < 1$ by $|\phi'| < k$ where $0 < k < 2$, the inequality would become $$\left|\int_0^\pi e^{i\phi(x)}\ dx \right| > \frac{2}{k} \sin\left(\frac{k \pi}{2}\right)$$