Comparable positive functions

Positive functions $f$ and $g$ are comparable, written $f\asymp g$ , on a specified domain if constants $0<c\le C<\infty$ satisfy

cg\le f\le Cg.

For an asymptotic domain these inequalities need only hold eventually. Equivalently, both $f=O(g)$ and $g=O(f)$ , with the implicit constants independent of the variables declared uniform.

Consequences

For fixed real $a$ , comparability implies $f^a\asymp g^a$ ; negative powers reverse the individual inequalities. Comparability does not say that $f/g$ has a limit.