Finding bounds on $g$ if $g(\frac{n}{2})=n\frac{m}{2}-\left[\lceil \frac{nm}{m+1}\rceil-\lceil\frac1{2}\lceil\frac{n}{m}\rceil\right]m$

Question

If $g,n,m\in\mathbb{N}$ and $n$ be even with $m\le n-2$ and $$g(\frac{n}{2})=n\frac{m}{2}-\left[\displaystyle\lceil \frac{nm}{2(m+1)}\displaystyle\rceil-\displaystyle\lceil\frac1{2}\displaystyle\lceil\frac{n}{m}\rceil\rceil\right]m$$ then what would be the best upper bound on $g$?

After using the elementary bounds $\lceil x\rceil\le x+1$, I got $g\le m-\frac{m^2}{m+1}+1+\frac{3m}{n}$, but I dont think this the best bound.Similarly, I think the lower bound is $1$. Any hints on how to solve? Thanks beforehand.

Answer 1

Since $n$ is even, I think it looks simpler if you replace it with $k = 2n$

Then you have:

$g,n,k\in\mathbb{N}$ and $m\le 2k-2$ and $$g(n)=km-\left[\displaystyle\lceil \frac{km}{(m+1)}\displaystyle\rceil-\displaystyle\lceil\frac1{2}\displaystyle\lceil\frac{2k}{m}\rceil\rceil\right]m$$

I don't have much time at the moment, and I will revisit this later. I'd use a "comment", but I don't have enough reputation yet...