Suppose \(f\) is entire and for every \(z\), \[ \abs{f(z)}\leq \abs{\sin(z)} .\] Characterize all possibilities for \(f\).
Solution
Write \(g(z) \da f(z) / \sin(z)\), which is meromorphic with singularities at the zeros of \(\sin(z)\) and bounded by 1. By boundedness, these singularities are removable, so \(g\) extends to a bounded entire and thus constant function. So \(f(z) = c\sin(z)\) where \(\abs{c} \leq 1\).