  Let $A$ be a complex unital Banach algebra. Using a connection between the
spectral distance and the growth characteristics of a certain entire map into
$A$, we derive a generalization of Gelfand's famous Power Boundedness Theorem.
Elaborating on these ideas, with the help of a Phragm\'{e}n-Lindel\"{o}f device
for subharmonic functions, it is then shown, as the main result, that two
normal elements of a $C^*$-algebra are equal if and only if they are
quasinilpotent equivalent.
