A reverse Denjoy theorem. II (Q977304)

For \(r\geq 0\) and \(\alpha\in (0,\pi)\), let \(C_1= \{z: z= re^{i(\pi-\alpha)}\}\) and \(C_2= \{z: z= re^{i(\pi+\alpha)}\}\) be rays from the origin, and let \(D= \{z:|\arg z-\pi|< \alpha\}\). Let \(u\) be a non-constant subharmonic function in \(\mathbb{C}\), with \(B(r, u)= \sup\{u(z):|z|= r\}\) and \(A_D(r, u)= \text{inf}\{u(z): z\in\overline D_r\}\), where \(D_r= D\cap\{|z|= r\}\). Then the authors show that the lower order of \(u\) is at least \(\pi/(2\alpha)\), if \(u(z)= \big(1+ o(1)\big)B\big(|z|,u\big)\) as \(z\to\infty\) on \(C_1\cup C_2\) and \(A_D(r,u)= o\big(B(r,u)\big)\). This result is closely related to an earlier result of the authors [Bull. Lond. Math. Soc. 41, No. 1, 27--35 (2009; Zbl 1167.31002)], in which (1) \(C_1\), \(C_2\) are simple non-intersecting curves joining \(0\) to \(\infty\), enclosing a domain \(D\) with \(\text{meas}\big\{\arg z: z\in\overline D_r\big\}\leq 2\alpha\), \(0<\alpha< \pi\), for all large \(z\in C_1\cup C_2\), where \(\overline D_r=\overline D\cap\big\{|z|= r\big\}\), (2) \(u(z)= B(r,u)\) for all large \(z\in C_1\cup C_2\), and (3) \(A_D(r,u)= o\big(B(r,u)\big)\).