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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 137, Issue 3
  • June 2007, pp. 457-474

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)
  • DOI: http://dx.doi.org/10.1017/S0308210506000102
  • Published online: 27 June 2008
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.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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