Inequality with bounded functions

Question

Consider

$$8-\frac{8B}{A}+\frac{2B(B-1)}{(A+1)(A-1)}$$

I would like to check if this expression is positive or not.

I know that $a \leq A \leq b$, for some positive integers $a,b$. Moreover, I know that $B\geq c$, another positive integer.

How would you show that?

I have been trying to separate $A$ and $B$ in order to use their boundness. But I am super stuck here.

Answer 1

I think it depends on the values of $A$ and $B$.

The numerator evaluates to $8(A-B)(A^2-1)+2AB(B-1)$. Using the fact that both $A$ and $B$ are $\geq 1$, the numerator can only be negative when $B>A$. My rough algebraic manipulation seems to suggest that the numerator is positive when $B>4A+1$. When $A<B<4A+1$, however, it is possible for the numerator to be negative. I tried $B=2A$ and found that the expression will be negative in that case if $A>2$.

Answer 2

Note that your term simplifies to $$2\,{\frac {4\,{A}^{3}-4\,{A}^{2}B+{B}^{2}A-BA-4\,A+4\,B}{A \left( A+1 \right) \left( A-1 \right) }} $$