Non-decreasing Functions, Exceptional Sets and Generalised Borel Lemmas

The Borel lemma says that any positive non-decreasing continuous function T satisfies T (r+1/T (r)) ≤ 2T (r) outside of a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. They appear also when using the existence of sufficiently many finite-order meromorphic functions as a criterion to detect difference equations of Painlevé type. In this paper, we consider generalisations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.