I have a function $f(n)$ defined over $n \in \mathbb{N}$ which I would like to bound asymptotically. I have proven that $f(a) \leq g(a)$ for some subset of naturals $a \in A \subset \mathbb{N}$ and I know $g(n) \sim h(n)$. The function $g(n)$ is rather ugly, but $h(n)$ is simple, so I would prefer to relate the growth of $f$ to $h$ instead of $g$. If I could show $f$ was monotone, (I think) that would be enough to claim $f(n) \sim \textrm{O}(h(n))$, but for the time being I am unable to show that.
My question is: is there a meaningful way to denote this relationship? I initially tried saying $f(n) \leq h(n)$ for infinitely many $n \in \mathbb{N}$, but we can't even say that because we only know: $$\lim_{n \to \infty} \frac{g(n)}{h(n)}=1$$ Maybe I'm looking for something like $f(n) \lesssim h(n)$, but I do not know exactly what this symbol means. Does anyone know how I should describe this relation?
Edit: I've done some thinking about the examples provided in the comments and I think I thought of a reasonable way to describe this relation. The most "meaningful" description I can use under the given information is that $f(n) \neq \omega(h(n))$. This is a slightly weaker claim than $f(n) = \textrm{O}(h(n))$, which I do not have enough information to claim as is, but I think this is still a nice characterization using asymptotic notation.