Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations
Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations
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DOI：
10.1017/s0308210506000102
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发表时间：
2007-06
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影响因子：
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通讯作者：
D. C. Barnett;R. Halburd;W. Morgan;R. Korhonen
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文献类型：
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作者：
D. C. Barnett;R. Halburd;W. Morgan;R. Korhonen
证明了，如果$f$是一个零阶亚纯函数，且$q\in\mathbb{C}$，则对于对数密度集$1$上的所有$r$，\开始{equation} \label{abstid} m\displaystyle（r，\frac{f（qz）}{f（z）}\displaystyle）=o（T（r，f））\tag{\ddag} \end{equation}.本文的其余部分由应用程序的身份\eqref{abstid}的研究值分布的零阶亚纯函数，特别是零阶亚纯解的$q$-差分方程。得到的结果包括$q$-移位类似的Nevanlinna理论的第二个主要定理，皮卡德定理，和克吕尼和Mohon'ko引理。
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.