Inequality involving abs. value of complex-valued multiple integral

1. Mar 10, 2006

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.

Last edited: Mar 10, 2006
2. Mar 10, 2006

shmoe

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

3. Mar 10, 2006

benorin

I got it. Thanks shmoe.