The max of the modulus of difference of a continuous function

Question

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 $$ \max_{x,y \in I} |f(x)-f(y)| = |f(a)-f(b)| $$

Answer 1

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, $$ |f(1/4)-f(0)|=7/4.$$

Answer 2

No. Set $$ f: [0,2] \to \mathbf{C}: x \mapsto \begin{cases}(1-x)+i \,\,& \text{ if }x \in [0,1]\\ \frac{x-1}{2}+i\,\frac{3-x}{2} & \text{ if }x \in [1,2]. \end{cases} $$