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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.

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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 $\mathbb C$.

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