I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function:
$\mathcal{f}: \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow\lbrack0,1]$ such that $t\geq0$
$ \mathcal{f}\left( u, d, t\right) =\left\{ \begin{array} [c]{lll} 0 & & t=0\\ \frac{u}{u+d+\frac{1}{t}} & & \text{otherwise} \end{array} \right. $
where $\mathcal{t}$ stands for the max value of $(\mathcal{u} + \mathcal{d})$ amongst the entities
I conjecture that the function satisfies the following properties:
- For two tuples $(\mathcal{u}_1, \mathcal{d}_1)$ and $(\mathcal{u}_2, \mathcal{d}_2)$, if $\frac{\mathcal{u}_1} {\mathcal{u_1} + \mathcal{d_1}} \ge \frac{\mathcal{u}_2} {\mathcal{u_2} + \mathcal{d_2}}$ then $\mathcal{f}(u_1, d_1, t) \ge \mathcal{f}(u_2, d_2, t)$.
As an example for $(\mathcal{u}_1, \mathcal{d}_1) = (10,0)$ and $(\mathcal{u}_2, \mathcal{d}_2) = (999,1)$, $\mathcal{t} = 1000$ and $\mathcal{f}(u_1, d_1, t) = 0.9999$ and $\mathcal{f}(u_2, d_2, t) = 0.9989$.
- For two tuples $(\mathcal{u}_1, \mathcal{d}_1)$ and $(\mathcal{u}_2, \mathcal{d}_2)$, if $\frac{\mathcal{u}_1} {\mathcal{u_1} + \mathcal{d_1}} = \frac{\mathcal{u}_2} {\mathcal{u_2} + \mathcal{d_2}}$ and $( \mathcal{u_1} + \mathcal{d_1}) \ge ( \mathcal{u_2} + \mathcal{d_2})$then $\mathcal{f}(u_1, d_1, t) \ge \mathcal{f}(u_2, d_2, t)$.
As an example for $(\mathcal{u}_1, \mathcal{d}_1) = (45,5)$ and $(\mathcal{u}_2, \mathcal{d}_2) = (9,1)$, $\mathcal{t} = 50$ and $\mathcal{f}(u_1, d_1, t) = 0.8996$ and $\mathcal{f}(u_2, d_2, t) = 0.8982$.
Question: As the examples also confirm, I highly believe that the properties hold for the defined function but I'm yet to prove them. I'll appreciate any help on this manner.