The max of the modulus of difference of a continuous function Let $I=[a,b]$ be a closed real interval
Let $f: I \to \mathbb{C}$ be a continuous function such that $|f(x)|$ is strictly decreasing
I would like to know if is it true that
$$
\max_{x,y \in I}
|f(x)-f(y)|
=
|f(a)-f(b)|
$$

This is not true, consider the spiral 
$$f(t):=(1-t)e^{4\pi it}$$
with $t\in[0,1/2]$. The modulus is of course decreasing and we have
$$
|f(1/2)-f(0)|=1/2.
$$ 
However,
$$
|f(1/4)-f(0)|=7/4.$$
