- holomorphic with respect to the variable and measurable with respect to the variable
- such that there is a function with for any and . (this is a "domination" of )
Then the function is holomorphic, and .
(I didn't specify the intervals or domains, this is whatever you want)
This theorem is quite powerful. It shows for instance that the "integral function" is indefinitely differentiable, using only a domination of the function (and not of its derivatives).
In your situation, you'll probably have to apply this theorem to compact subsets of the domain (otherwise you won't be able to get the domination).