Properties of Ahlfors constant in Ahlfors covering surface theory

This paper is a subsequent work of [Invent. Math., 2013, 191: 197-253]. The second fundamental theorem in Ahlfors covering surface theory is that, for each set E_q of q (≥3) distinct points in the extended complex plane \overline{ℂ}; there is a minimal positive constant H₀ (E_q) (called Ahlfors constant with respect to E_q), such that the inequality(q-2)A(\sum )-4\pi ♯(f^{-1}(E_q)\cap U)\le H_0(E_q)L(\partial \sum )holds for any simply-connected surface{\displaystyle \sum \mathrm{=}（f,\bar{U}）} ; where A(\sum) is the area of\sum; L(\partial \sum) is the perimeter of\sum; and # denotes the cardinality. It is difficult to compute H₀ (E_q) explicitly for general set E_q; and only a few properties of H₀(E_q) are known. The goals of this paper are to prove the continuity and differentiability of H₀ (E_q); to estimate H₀ (E_q); and to discuss the minimum of H₀ (E_q) for fiixed q.