Proving integer inequality I am having trouble proving this inequality:
For positive integers $p$, $q$, $c$, $k$ such that $2 \le p < q$:
$$
\frac{(p-1) \, p^{k-1} / \, (c p^k - 1)}{(q-1) \, q^{k-1} / \, (c q^k - 1)} \le 1
$$
Note that I have verified it empirically in code for thousands of small values.

Your statement is equivalent to saying the function
$$f_{c,k}(x)= \frac{(x-1) \, x^{k-1} }{ c x^k - 1} $$
is increasing on the set of integers $x \ge 2$. 
Now, extend this function to the real interval $(1, \infty)$.
You have
$$ f'(x)=\frac{x^{k-2} }{ (c x^k - 1)^2} (cx^k-kx+k-1)$$
we clearly have
$$\frac{x^{k-2} }{ (c x^k - 1)^2}  >0$$
and by Bernoulli's inequality
$$x^k \geq 1+k(x-1)=1+kx-k$$
thus
$$cx^k-kx+k-1 \geq x^k-kx+k-1 \geq 1+kx-k-kx+k-1=0$$
This shows that $f(x)$ is increasing on $(1, \infty)$, which proves your statement.
