= Solution
The <complex exponential function> $E(z)=e^z$ maps the strip bijectively and holomorphically to the <upper half-plane>: its modulus is $e^{\Re z}>0$ and its argument is $\Im z\in(0,\pi)$. Its inverse is the <holomorphic> <logarithm> $L(w)=\log|w|+i\operatorname{Arg}w$ with $\operatorname{Arg}w\in(0,\pi)$.
Part (iii) applies to both maps. The required inverse correspondences are
$$
\boxed{h\in\mathcal H(\Omega_2)\longmapsto h\circ E\in\mathcal H(\Omega_1),\qquad k\longmapsto k\circ L.}
$$
They compose to the identity because $L\circ E$ and $E\circ L$ do. This proves injectivity and surjectivity, not merely a map between the two sets.
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