Find all entire functions \(f\) satisfying \[ \abs{f(z)} \geq \abs{z} + 1 &&\forall z\in \CC .\]
Solution
The inequality implies \(f\) has no zeros, so \(g(z) \da 1/f(z)\) is entire. Moreover it is bounded on \(\CC\), since \[ \abs{g(z)} \leq {1\over \abs{z} + 1} \leq 1 ,\] so \(g\equiv c\) is constant by Liouville. Since \(\abs{g(z)}\le 1/(\abs z+1)\to0\) as \(\abs z\to\infty\), \(c=0\), which contradicts \(g=1/f\) having no zeros. So there are no such entire functions.