I just want to make a clarification with regard to bounded meromorphic functions on the complex plane $\mathbb{C}$. Would they be constant?

Here's what I do know:
$(1)$ Liouville's Theorem states that a bounded $\textit{entire}$ function (holomorphic on $\mathbb{C}$) is constant.
$(2)$ Holomorphic functions on all of the extended complex plane $\mathbb{C} \cup \{\infty \}$ are constant.


Yes: if a meromorphic function is bounded, it is in particular bounded near its singularities. This means that the singularities are removable, so the function extends to a bounded holomorphic function on all of $\mathbb C$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.