Nevanlinna theory of the Hahn difference operators and its applications

The version of Nevanlinna theory based on Hahn difference operators Dq,c(g)=g(qz+c)−g(z)(q−1)z+cfor meromorphic function g of zero order is studied in this paper. Firstly, the logarithmic derivative lemma for the Hahn difference operators is obtained. As applications of the logarithmic derivative lemma, the Clunie lemma for the Hahn difference operators and the growth of solutions of Hahn difference equations are considered, respectively. Secondly, the second fundamental theorem for the Hahn difference operators is built which is used to prove the deficiency relation, Picard's theorem, and the five-value theorem in the sense of Hahn difference operators. Finally, the existence of solutions of Fermat-type Hahn difference equations is also discussed.