If I have a pair of functions increasing together or decreasing together, I certainly have:
\begin{equation} min(|\Delta F|,|\Delta G|) > 0 \iff \frac{\Delta F}{\Delta G} > 0 \end{equation}
I am tempted to express this relationship as follows:
\begin{equation} \forall x \in \mathbb{R},F(x) \propto G(x) \end{equation}
However, $\propto$ is not quite the appropriate symbol as I'm trying to describe variables which aren't necessarily proportional to each other. This problem, which might seem contrived, actually occurred in the context where I'm trying to calculate:
\begin{equation} \max_{x} F(x) =\int_{t=0}^x f(t) dt \end{equation}
However, $F(x)$ is an intractable expression whereas I found a different function $G$ such that $G(x)$ is tractable so instead I calculate:
\begin{equation} \max_{x} G(x) \end{equation}
I noticed that this kind of trick regularly appears for the types of problems I'm trying to solve, so I think such a symbol must already exist.
Note: I am still wondering whether $\propto$ might be sufficient. I might be overcomplicating this.
I do not think that there is any special symbol for this relationship between two functions of real numbers, so perhaps it is best to pick some little-used symbol and define it to mean that in your context.
For example:
If $F,G: \mathbb{R}\to\mathbb{R}$ then $F\bowtie G$ means that if $a,b\in\mathbb{R}$ and $a\le b$ then $F(a)\le F(b)\iff G(a)\le G(b)$ and $F(a)\ge F(b)\iff G(a)\ge G(b)$.