Non constant analytic function from $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}.$ Does there is non constant analytic function from  $\{z\in\mathbb{C}:z\neq 0\}$ to $\{z\in\mathbb{C}:|z|>1\}?$ According to me there is no such  non constant analytic function because if there is any such function say $f,$ then $f$ can have  either a pole or essential singularity at $z=0$. In the case of pole Picard's theorem of meromoprphic function will work and in the case of essential singularity we know that image of  any neighbourhood of essential singularity  is dense in $\mathbb{C}$, so in both of the cases we get a contradiction. So no such non constant analytic function. Am i right? Please suggest me. Thanks.

If so then $1/f$ is bounded. Hence $1/f$ has a removable singularity at the origin, giving a bounded entire function.
