Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ and assuming also that $$\frac{1}{c_{m+n+k}}=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}f\left(z\right)z^{m+n+k}dz $$ where $a>0 $ and $f$ is an holomorphic function. The question is:
under which hypothesis can we exchange the integral with the double series?
I know that if we consider $$\sum_{n\geq1}\left|b_{n}\right|\int_{a-i\infty}^{a+i\infty}\left|f\left(z\right)\right|\left|z^{m+n+k}\right|\sum_{m\geq1}\left|a_{m}\right|\left|dz\right|\tag{2} $$ and $$\int_{a-i\infty}^{a+i\infty}\left|f\left(z\right)\right|\sum_{n\geq1}\sum_{m\geq1}\left|a_{m}b_{n}z^{m+n+k}\right|\left|dz\right|\tag{3} $$ and we prove that both converge, we can exchange. This problem comes out from a work of mine. I have to find the smallest $k $ such that we can exchange the summation and the integral signs. If I use $(2)$ and $(3)$ I get $k>1 $ and this result is optimal, but I would like to keep $k>1/2 $. Does there exist another way to prove the exchange?