I find the following interesting fact when I study the characteristic functions in probability theory.
For any $x\in \mathbb{R}$, the following inequality holds
$$ \min_{t\in [1,5]}|1-e^{itx}|\le 1. $$
Proof: if $|x|\le \frac{\pi}{3}$, take $t=1$, the inequality holds; if $|x|>\frac{\pi}{3}$, then the range of the argument of $e^{itx}$ (which is $4|x|>\frac{4\pi}{3}$) will intersect with $[-\frac{\pi}{3},\frac{\pi}{3}]$ on the circle, the conclusion also holds.
I'm curious that whether the following generalized version is true:
$$ \min_{t\in [1,5]} |1-\psi(t)|<1, $$
where $\psi(t)=\mathbb{E}_{p}[e^{itx}]$ is any characteristic function of probability measure $p$ on $\mathbb{R}$.
The function $\psi(t)=(1-t^2)\exp(-t^2/2)$ is the characteristic function of a random variable with density function $z^2\exp(-z^2/2)/\sqrt{2\pi}$. Similarly, $\psi(2t)$ is also a characteristic function. But $\psi(t)<0$ for all $t>1$, and so $\min_{t\in[1,5]}|1-\psi(2t)|>1$.