I have problems with this question. It is as follows:
Let $g:A\times\Omega\rightarrow\mathbb{C}$, $\ A\subset\mathbb{R}^n$, and an open set $\Omega\subset\mathbb{C}$. The function $g$ satisfies the following properties:
- For each $x\in A$, $\ g(x,z)=g_x(z)$, $\ g_x:\Omega\rightarrow\mathbb{C}$, belongs to $\mathcal{H}(\Omega)$.
- Exists a function $h:A\rightarrow\mathbb{R}$, with $h\in L^1(A)$, such that for every $x\in A$, $z\in\Omega$, $\lvert\frac{\partial g}{\partial z}(x,z)\lvert\leq h(x)$
I have to show that the functions defined as $$f(z)=\int_Ag(x,z)\ dx$$ belongs to $\mathcal{H}(\Omega)$, and $$f'(z)=\int_A\frac{\partial g}{\partial z}(x,z)\ dx$$
My guess:
I'm (almost completely) lost. In the first statement (belongs to $\mathcal{H}(\Omega)$), it is straight that any function function with belongs to $\mathcal{H}(\Omega)$, its integral belongs to $\mathcal{H}(\Omega)$ too? I don't think so, but I don't know how to show it. And in the second statement, maybe I have to delimit the sequence of $g_x$ to put, inside the integral sign, the differentiation?
