Conformal-onto vs Onto map.

Question

In Balazard, Saias and Yor paper they assume that f is in the Hardy space $ H^p\mathbb{(D)}$ where $\mathbb{D}=\{z\in \mathbb{C}\mid |z|<1\}$. Let, $f^*$ denote the function defined almost everywhere on the unit circle $\partial\mathbb{D}=\{z\in \mathbb{C}\mid |z|=1\}$ by $$f^*(e^{i\theta})= \lim_{r\to 1^-}f(re^{i\theta})$$ Then they write $$s=s(z)=\frac{1}{1-z} $$ Then the above formula is a Conformal map from the disc $\mathbb{D}$ onto the semi plane $\Re(s)>1/2$. Later they use Jensens formula and Hardy spaces properties to prove $$\frac{1}{2\pi}\int_{-\pi}^\pi \log|f^*(e^{i\theta})|d\theta=\log|f(0)|+\sum_{|\alpha|<1,f(\alpha)=0}\log\frac{1}{|\alpha|} $$ Finally define $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta$ is the Riemann zeta function then, $$\sum_{|\alpha|<1,f(\alpha)=0}\log\frac{1}{|\alpha|}= \sum_{\Re(\rho)>1/2}\log \left|\frac{\rho}{1-\rho}\right|$$ Question Why in mapping the Disc onto the semi plane $\Re(s)>1/2$ we need a Conformal map? Can't we just have an onto map? Please answer.