Research Article On the Iterated Exponent of Convergence of Solutions of Linear Differential Equations with Entire and Meromorphic Coefficients

Abstract

In this paper, a meromorphic function will mean meromorphic in the whole complex plane. We shall use the standard notations in Nevanlinna value distribution of meromorphic functions [1, 2] such as 𝑇𝑇 (𝑇𝑇𝑧 𝑓𝑓𝑘, 𝑁𝑁 (𝑇𝑇𝑧 𝑓𝑓𝑘, 𝑚𝑚 (𝑇𝑇𝑧 𝑓𝑓𝑘. For the definition of the iterated order of a meromorphic function, we use the same definition as in [3],[4, p. 317],[2, p. 129]. For all 𝑇𝑇 𝑟 𝑟, we define exp 𝑘𝑇𝑇 𝑟𝑧 𝑟𝑟𝑇𝑇 and exp 𝑝𝑝+ 𝑘𝑇𝑇 𝑟𝑧 exp (exp𝑝𝑝𝑇𝑇𝑘, 𝑝𝑝 𝑟 𝑝. We also define for all 𝑇𝑇 sufficiently large log𝑘𝑇𝑇 𝑟𝑧 log 𝑇𝑇 and log 𝑝𝑝+ 𝑘𝑇𝑇 𝑟𝑧 log (log𝑝𝑝𝑇𝑇𝑘𝑧 𝑝𝑝 𝑟 𝑝. Moreover, we denote by exp0𝑇𝑇 𝑟𝑧 𝑇𝑇, log0𝑇𝑇 𝑟𝑧 𝑇𝑇, log𝑘𝑘𝑇𝑇 𝑟𝑧 exp𝑘𝑇𝑇 and exp𝑘𝑘𝑇𝑇 𝑟𝑧 log𝑘𝑇𝑇.