Bounded holomorphic function on half-plane

Question

Let $f$ be a holomorphic function and bounded of half-plane $\mathbb H=\{z=x+iy\in \mathbb C| y>0\}$. I consider the function $G(z)=\int_0^1 f(tz)dt$. I think $G(z)$ must be holomorphic on $\mathbb H$, but I have no idea to prove it. Could you give me some hints.

Answer 1

A standard way to show that such integrals are holomorphic is to use the dominated convergence theorem to see that $G$ is a continuous function, and then use Morera's theorem to see it is holomorphic.

Since $(t,z) \mapsto f(tz)$ is continuous on $(0,1]\times \mathbb{H}$ and $f$ is bounded, the application of the dominated convergence theorem is straightforward here.