Proceedings of the Royal Society of Edinburgh: Section A Mathematics



Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations


D. C. Barnett a1, R. G. Halburd a1, W. Morgan a1 and R. J. Korhonen a2
a1 Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK (d.c.barnett@lboro.ac.uk; r.g.halburd@lboro.ac.uk; w.morgan@lboro.ac.uk)
a2 Department of Mathematics, University of Joensuu, PO Box 111, 80101 Joensuu, Finland (risto.korhonen@joensuu.fi)

Article author query
barnett dc   [Google Scholar] 
halburd rg   [Google Scholar] 
morgan w   [Google Scholar] 
korhonen rj   [Google Scholar] 
 

Abstract

It is shown that, if $f$ is a meromorphic function of order zero and $q\in\mathbb{C}$, then

\begin{equation} \label{abstid} m\bigg(r,\frac{f(qz)}{f(z)}\bigg)=o(T(r,f)) \tag{\ddag} \end{equation}

for all $r$ on a set of logarithmic density $1$. The remainder of the paper consists of applications of identity \eqref{abstid} to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of $q$-difference equations. The results obtained include $q$-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.

(Published Online June 27 2008)
(Received January 13 2006)
(Accepted April 20 2006)