q-DIFFERENCE ANALOGUE OF THE LEMMA ON THE LOGARITHMIC DERIVATIVE AND SOME APPLICATIONS

Abstract

Recently, Nevanlinna theory applied to study difference-differential equations, also value distribution of difference-differential polynomials. This research direction has attracted the attention of many mathematicians in the country as well as around the world. In this paper, by using q-difference analogue of the lemma on the logarithmic derivative and Nevanlinna theory for meromorphic functions in several variables, we study the proximity function of solutions to q-shift difference-partial differential. Our results show that under some suitable conditions of degree of equations, proximity function of solutions is small function in comparing with characteristic functions. In addition, we establish a new lemma on the counting function of zeros of the partial derivative of meromorphic function in several variables, and apply that result to study the value distribution of difference-partial differential polynomials. In our best knowledge, our results are new and some future works can be done by using our previous results.