Lower bound of admissible functions on sphere (Q1865019)

Let \(S^2=\mathbb{C} P^1\) be the complex projective line and let \(\psi( [z_0,z_1])= \log{|z_0|^2 |z_1|^2 \over(|z_0 |^2 +|z_1|^2)}\). This function has the maximal value \(\log 4\) and tends to minus infinity if \(z_0\) or \(z_1\) goes to zero. The main result of this paper is that \(\psi\) is a lower bound for a class of functions called admissible.