Abstract
For α satisfying 0 < α < π, suppose that C 1 and C 2 are rays from the origin, C 1: z = re i(π−α) and C 2: z = re i(π+α), r ≥ 0, and that D = {z: | arg z − π| < α}. Let u be a nonconstant subharmonic function in the plane and define B(r, u) = sup|z|=r u(z) and A D (r, u) = \( \inf _{z \in \bar D_r } \) u(z), where D r = {z: z ∈ D and |z| = r}. If u(z) = (1 + o(1))B(|z|, u) as z → ∞ on C 1 ∪ C 2 and A D (r, u) = o(B(r, u)) as r → ∞, then the lower order of u is at least π/(2α).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. M. Anderson and A. Baernstein, The size of the set on which a meromorphic function is large, Proc. London Math. Soc. (3) 36 (1978), 518–539.
A. Baernstein, Proof of Edrei’s spread conjecture, Proc. London Math. Soc. (3) 26 (1973), 418–434.
M. R. Essén, The cos πλ theorem, Lecture Notes in Mathematics, Vol. 467, Springer-Verlag, Berlin, 1975.
P. C. Fenton and J. Rossi, A reverse Denjoy theorem, Bull. LondonMath. Soc. 41 (2009), 27–35.
W. K. Hayman, Subharmonic Functions, Vol. 1, Academic Press, New York, 1976.
W. K. Hayman, Subharmonic Functions Vol. 2, Academic Press, New York, 1989.
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author was supported in part by the Department of Mathematics and Statistics at the University of Otago. He would like to thank them for their hospitality.
Rights and permissions
About this article
Cite this article
Fenton, P.C., Rossi, J. A reverse Denjoy theorem II. JAMA 110, 385–395 (2010). https://doi.org/10.1007/s11854-010-00010-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-010-00010-7