On an ultrahyperelliptic surface with Picard constant three (Q1321779)

Let \(S\) be an open Riemann surface, \(M(S)\) the family of nonconstant meromorphic functions and \(P(f)\) the number of values which are not taken by \(f \in M(S)\). The Picard constant \(P(S)\) of \(S\) is defined by \(P(S) = \{P(f) : f \in M (S)\}\). Let \(R\) be the ultrahyperelliptic surface defined by \(y^2 = G(z)\), where \(G(z) = 1 - 2 \beta_1 e^{H(z)} - 2 \beta_2 e^{L(z)} + \beta^2_1 e^{2H (z)} - 2 \beta_1 \beta_2 e^{H (z) + L(z)} + \beta^2_2 e^{2L(z)}\) with two nonconstant entire functions \(H\) and \(L\) \((H(0) = L(0) = 0)\) and two non- zero constants \(\beta_1\) and \(\beta_2\). Then we know that \(3 \leq P (R) < 4\) and \(P(R) = 4\) in the following four cases: (i) \(H = L\); (ii) \(H = 2L\), \(16 \beta_1 = \beta^2_2\); (iii) \(2H = L\), \(\beta^2_1 = 16 \beta_2\); (iv) \(H = - L\), \(16 \beta_1 \beta_2 = 1\). Is \(P(R) = 3\) right with the above four exceptional cases? \textit{M. Ozawa} [Kōdai Math. Sem. Rep. 19, 245-256 (1967; Zbl 0164.088)] proved that this is valid when \(H\) and \(L\) are polynomials. The author proves that this statement is true when \(L = \lambda H + K\), where \(\lambda\) is a rational number and \(K\) is an entire function satisfying \(m(r,e^K) = o(m(r,e^H))\), \(r \to \infty\), outside a set of \(r\) of finite measure.