Let h be the holomorphic function $h(z)=\int_0^{+\infty}e^{tz}t^{-t}dt$.
I am trying to show that $g(z):=h(z-2i\pi)$ is bounded on every line passing by the origin.
I don't understand the difference between $g$ being bounded on all lines passing by the origin and the whole $\mathbb{C}$ plane ? Isn't $\mathbb{C}$ the union of all lines passing by the origin?
What I did: I showed that h is bounded on $\{z\in \mathbb{C}: |\Im(z)|\ge\pi\}$
I don't know how to go from there? Many thanks!