# Notation for asymptotic bounds with coefficients?

I ran into the following notation issue:

Suppose that I have two functions $f$ and $g$ and I know that $\displaystyle \lim_{n \rightarrow \infty} \frac{f(n)}{g(n)} = c$ for some $1 < c < \infty$.

I want to express that $f$ is an asymptotic upper bound for $g$, but saying $g(n) = O(f(n))$ is worthless because coefficients are dropped.

Is there a way to express this upper bound more tersely than the limit expression above? In other words is there any "order-O-like" notation for expressing this?

Clarification: I'm not looking for any of the symbols here. The question may be too pedantic.

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I know that in number theory we usually say that $f(n) << g(n)$ (call it "$f$ is less-than-less-than $g$") when $f$ is asymptotic to $g$ multiplied by some undetermined positive constant, but I don't know if this is what you're looking for. (Here $f$ and $g$ are positive functions of reals) –  Patrick Da Silva Feb 2 '12 at 9:52
It is not clear what you want, since $f(n)\sim cg(n)$ for some $c>1$ is precisely equivalent to the given limit. What additional information do you intend is an asymptotic upper bound to convey? –  Brian M. Scott Feb 2 '12 at 16:20
@Patrick Da Silva: that's exactly what I meant; I haven't seen that before. –  Huck Bennett Feb 2 '12 at 18:03
@Brian Scott: It's just a question of notation. The point is "$f(n) << g(n)$" is more concise than "$f(n) \sim cg(n)$ for some $c > 1$"; in particular it doesn't introduce another variable to describe the relationship between $f$ and $g$. –  Huck Bennett Feb 2 '12 at 18:06
A widely understood notation seems distinctly preferable to a very slightly more concise notation that needs to be explained. –  Brian M. Scott Feb 2 '12 at 18:14

I know that in number theory we usually say that $f(n) \ll g(n)$ (call it "$f$ is less-than-less-than $g$") when $f$ is asymptotic to $g$ multiplied by some undetermined positive constant, but I don't know if this is what you're looking for. (Here $f$ and $g$ are positive functions of reals)

Hope that helps,

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