International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 537478, 24 pages
doi:10.1155/2011/537478
Research Article

Value Distribution for a Class of Small Functions in the Unit Disk

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA

Received 20 October 2010; Accepted 21 January 2011

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2011 Paul A. Gunsul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

If 𝑓 is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function 𝑇 ( 𝑟 , 𝑓 ) could be used to categorize 𝑓 according to its rate of growth as | 𝑧 | = 𝑟 . Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer 𝑘 , 𝑚 ( 𝑟 , 𝑓 ( 𝑘 ) / 𝑓 ) = 𝑜 ( 𝑇 ( 𝑟 , 𝑓 ) ) as 𝑟 , possibly outside a set of finite measure where 𝑚 denotes the proximity function of Nevanlinna theory. If 𝑓 is a meromorphic function in the unit disk 𝐷 = { 𝑧 | 𝑧 | < 1 } , analogous results to the previous equation exist when l i m s u p 𝑟 1 ( 𝑇 ( 𝑟 , 𝑓 ) / l o g ( 1 / ( 1 𝑟 ) ) ) = + . In this paper, we consider the class of meromorphic functions 𝒫 in 𝐷 for which l i m s u p 𝑟 1 ( 𝑇 ( 𝑟 , 𝑓 ) / l o g ( 1 / ( 1 𝑟 ) ) ) < , l i m 𝑟 1 𝑇 ( 𝑟 , 𝑓 ) = + , and 𝑚 ( 𝑟 , 𝑓 / 𝑓 ) = 𝑜 ( 𝑇 ( 𝑟 , 𝑓 ) ) as 𝑟 1 . We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which l i m s u p 𝑟 1 ( 𝑇 ( 𝑟 , 𝑓 ) / l o g ( 1 / ( 1 𝑟 ) ) ) = + holds. We also explore connections between the class 𝒫 and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem.