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I know the formula of conformal mapping between upper half plane $\mathbb H$ and unit disc $\mathbb D$, so it would be sufficient to find a conformal mapping between the region $\{z=x+iy: xy>2, x>0\}$ and $\mathbb H$.

I first attempted to rotate $S$ by $-45^{\circ}$ and apply Conformal mapping $z+\frac{1}{z}$, how to see the mapping to hyperbolas? , but this did not work.

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    http://math.stackexchange.com/questions/603142/conformal-mappings-from-xy-1-to-the-upper-half-plane – 2017-01-29
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    Maybe this will help? – 2017-01-29

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Simply $w=z^2$ maps $S$ onto $\mathbb{H}^\prime=\{w\mid \operatorname{Im}w>4\}$ , so $$w=z^2-4i$$ maps $S$ onto $\mathbb{H}$.

Of course there is an approach to obtain the mapping $w=z^2-4i$ using mapping properties of $z+\frac{1}{z}$, however, it is very complicated.

To use mapping properties of $z+\frac{1}{z}$ is an essential way if you want to obtain a mapping function of $\{z=x+iy\mid xy<2\}$ to $\mathbb{H}.$