A lemma on the difference quotients

First, we are concerned with a lemma on the difference quotients due to Halburd, Korhonen and Tohge. We show that for meromorphic functions whose deficiency is origin dependent the exceptional set associated with this lemma is of infinite linear measure. In particular, for such entire functions in this set there is an infinite sequence {r(n)} such that m(r(n), f(z c)/ f(z)) not equal o(T(r(n), f)) for all r(n). Then we extend this lemma to the case of meromorphic functions f(z) such that log T(r, f) <= ar/(log r)(2+nu), a, nu > 0, for all sufficiently large r, by using a new Borel type growth lemma. Second, we give a discrete version of this Borel type growth lemma and use it to provide an extension of Halburd's result on first order discrete equations of Malmquist type.