I have the following relationship:
Given $x,y \in \mathbb{C}$ and $c \in \mathbb{R}_{+}$,
we have $\left||x| - c\right| = \left|x - e^{j\angle x}c\right| < \left||x| - e^{j\angle y}c\right| $, where $y \neq x$.
Now can I say $\left|x - e^{j\angle x}c\right| < \left|x - e^{j\angle y}c\right|$ for all $y \neq x$, i.e, to say,
$\left|x - e^{j\angle y}c\right|$ is the upper bound for $\left||x| - c\right|$