Find all entire functions \(f\) that satisfy \[ \abs{f(z)} \geq e^{\abs{z}} && \forall z\in \CC .\]
Solution
Claim: there are no such functions. Consider \(g(z) \da f(z)/e^z\), which is entire since \(e^z\) is nonvanishing. Now \(g\) is entire and \(\abs{g} \geq 1\) everywhere, so \(\im(g) \intersect \DD\) is empty. This contradicts Casorati-Weierstrass, which requires that \(\im g\) be dense in \(\CC\).
Alternatively, note \(f\neq 0\) by the inequality, so \(1/f\) is bounded and entire and thus constant. However, \(f(z) = c\) contradicts the inequality, since \(e^{\abs{z}}\to \infty\) as \(\abs{z}\to \infty\).
Alternatively, note that \(e^{\abs{z}} \geq 1\) for all \(z\), so \(\im(f) \intersect \DD\) is empty, again contradicting Casorati-Weierstrass.