A reverse Denjoy theorem III

作者：Fenton P C; Rossi John^*

来源：Science China Mathematics, 2010, 53(3): 657-662.

DOI：10.1007/s11425-010-0043-5

• Denjoy conjecture
• subharmonic function
• tract

Suppose that C(1) and C(2) are two simple curves joining 0 to infinity, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, the set {theta : re(i theta) is an element of (D) over bar} has measure at most 2 alpha, where 0 < alpha < pi. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = Phi(vertical bar z vertical bar) for all large z is an element of C(1) boolean OR C(2) boolean OR similar to D, where Phi(vertical bar z vertical bar) is a convex, non-decreasing function of vertical bar z vertical bar and similar to D is the complement of D. Let A(D)(r, u) = inf{u(z) : z is an element of (D) over bar and vertical bar z vertical bar = r}. It is shown that if A(D)(r, u) = O(1) then lim inf(r-->infinity) B(r, u)/r(pi/(2 alpha)) > 0.

• 出版日期2010-3