- Find all meromorphic functions f on C such that
\[ | f ( z ) | \leq { \frac { \log ( 2 + | z | ^ { 2 } ) } { | z | } } \qquad { \mathrm { f o r ~ a l l ~ } } z \neq 0 . \]
Give explicit formulas for the functions and give a proof for your answer.
Solution
The functions are exactly \[ \boxed{f(z)=\frac{a}{z},\qquad a\in\mathbb C,\quad |a|\leq\log2,} \] viewed as meromorphic functions; \(a=0\) gives the identically zero function.
1. The function \(g(z)=zf(z)\) extends to an entire function.
At any nonzero point the given upper bound is locally finite. A pole there would make \(|f|\) unbounded in every punctured neighborhood, contradicting the bound. Thus \(f\) is holomorphic on \(\mathbb C\setminus\{0\}\). On that set, \[ |g(z)|\leq\log(2+|z|^2). \] This bounds \(g\) near zero, so the removable-singularity theorem gives an entire extension, still denoted \(g\) [@SS03]. Continuity preserves the bound at zero as well.
2. The entire extension is constant.
Write \(g(z)=\sum_{n\geq0}b_nz^n\). For every \(R>0\), Cauchy’s coefficient estimate gives \[ |b_n|\leq \frac{\log(2+R^2)}{R^n} \] [@SS03]. For any integer \(n\geq1\), the right-hand side tends to zero as \(R\to\infty\): for \(R\geq1\) its numerator is at most \(\log3+2\log R\), and \((\log R)/R^n\to0\). Hence \(b_n=0\) for every \(n\geq1\). Thus \(g=a\) is constant, and \(f(z)=a/z\) on the punctured plane, determining \(f\) as a meromorphic function.
3. The coefficient restriction is necessary and sufficient.
The original inequality for \(f=a/z\) is equivalent, after multiplication by \(|z|>0\), to \[ |a|\leq\log(2+|z|^2)\qquad(z\ne0). \] Letting \(|z|\downarrow0\) gives \(|a|\leq\log2\). Conversely, this last bound implies the displayed inequality for every nonzero \(z\), since the logarithm is increasing. Every such \(a/z\) is meromorphic on the plane, with at most a simple pole at zero. Therefore every stated coefficient is allowed, and no other function is allowed.