Suppose that \(g\) is entire and satisfies the functional equation \(g(1-z) = 1-g(z)\). Show that \(g(\CC) = \CC\).
The only constant solution of \(g(1-z)=1-g(z)\) is \(g\equiv1/2\), which is not surjective. The conclusion holds for every nonconstant solution.
Solution
Assume \(g\) is nonconstant; the remark treats constant \(g\). Suppose \(g\) omits a value \(a\). Setting \(z=1/2\) in the functional equation gives \(g(1/2)=1-g(1/2)\), so \(g(1/2)=1/2\) and \(a\neq 1/2\). The value \(1-a\) is also omitted: if \(g(z)=1-a\), then \(g(1-z)=1-g(z)=a\). Since \(a\ne1/2\), the values \(a\) and \(1-a\) are distinct, so the nonconstant entire function \(g\) omits two values, contradicting Picard’s little theorem.