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The complex exponential function gives the natural onto map:Its derivative is never zero, so it preserves angles locally. Conversely any point in this annulus has a logarithm with real part in $(1/4,1/2)$, proving surjectivity.

There is a terminology qualification. This map is not injective, since $e^{z+2\pi i}=e^z$ and the strip contains both points. If “conformal mapping onto” requires a biholomorphism, no such map exists: the strip is convex and simply connected space, whereas the annulus contains a loop of winding number one around zero that cannot contract inside it. A biholomorphism would be a homeomorphism and preserve this property.

Thus the complex exponential function is a locally conformal covering; a biholomorphism is impossible. It identifies imaginary translates by $2\pi i$. The printed request is satisfied under its local-angle-preserving convention, with the injectivity distinction made explicit. This is exponential covering of an annulus by a vertical strip.