Inequality involving abs. value of complex-valued multiple integral

[benorin]

How do I show that

$$0\leq \Re (s) +k\Rightarrow\left| (s)_k \int_0^1\cdots\int_0^1 (n+x_1+\cdots +x_k)^{-s-k}\, dx_1\cdots\, dx_k \right| \leq |(s)_k|n^{-\Re (s) -k}$$​

where k is a nongegative integer and $$(s)_k:=s(s+1)\cdots (s+k-1)$$ is the Pochammer symbol (aka the rising factorial) ?

If it helps, I know (and have previously proven) that

$$(s)_k\int_0^1\cdots\int_0^1 (n+x_1+\cdots +x_k)^{-s-k}\, dx_1\cdots\, dx_k = \sum_{m=0}^{k}(-1)^{m} \left(\begin{array}{c}k\\m\end{array}\right) (n+m)^{-s} =: \Delta ^k (n^{-s})$$​

where $$\Delta$$ is the forward difference operator.

 

[shmoe]

You don't need your previous work, just use the simplest upper bound for your integral you can think of.

 

[benorin]

I got it. Thanks shmoe.