A quantitative version of Picard's theorem (Q1815443)

Let \(f\) be an entire function with order at least 1/2. The author uses the main result of Ahlfors' theory of covering surfaces to prove \[ \lim_{r+ \infty} \sup i \biggl \{n(r,a)/ \bigl(\log M(r) \bigr) \biggr\} \geq 1/2 \pi \] where \(n(r,a)\) is the number of zeros of \(f(z)-a\) in \(\{z |z |\leq r\}\). Examples suggest that it should be possible to replace \(1/2\pi\) by \(1/ \pi\).