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This article is cited in 5 scientific papers (total in 5 papers)
Nevanlinna's five-value theorem for algebroid functions
Ashok Rathod Department of Mathematics,
Karnatak University,
Dharwad-580003, India
Abstract:
By using the second main theorem of the algebroid function, we study the following problem. Let $W_{1}(z)$ and $W_{2}(z)$ be two $\nu$-valued non-constant algebroid functions, $a_{j}\,(j=1,2,\ldots,q)$ be $q\geq 4\nu+1$ distinct complex numbers or $\infty$. Suppose that ${k_{1}\geq k_{2}\geq \ldots\geq k_{q},m}$ are positive integers or $\infty$, $1\leq m\leq q$ and $\delta_{j} \geq 0$, $j=1,2,\ldots,q$, are such that
\begin{equation*}
\left(1+\frac{1}{k_{m}}\right)\sum_{j=m}^{q}\frac{1}{1+k_{j}}+3\nu +\sum_{j=1}^{q}\delta_{j}<(q-m-1)\left(1+\frac{1}{k_{m}}\right)+m.
\end{equation*}
Let $B_{j}=\overline{E}_{k_{j}}(a_{j},f)\backslash\overline{E}_{k_{j}}(a_{j},g)$ for $j=1,2,\ldots,q.$ If
\begin{equation*}
\overline{N}_{B_{j}}(r,\frac{1}{W_{1}-a_{j}})\leq \delta_{j}T(r,W_{1})
\end{equation*}
and
\begin{equation*}
\liminf_{r\rightarrow \infty}^{}\frac{\sum\limits_{j=1}^{q} \overline{N}_{k_{j}}(r,\frac{1}{W_{1}-a_{j}})} {\sum\limits_{j=1}^{q}\overline{N}_{k_{j}}(r,\frac{1}{W_{2}-a_{j}})}> \frac{\nu k_{m}}{(1+k_{m})\sum\limits_{j=1}^{q} \frac{k_{j}}{k_{j}+1}-2\nu(1+k_{m}) +(m-2\nu-\sum\limits_{j=1}^{q}\delta_{j})k_{m}},
\end{equation*}
then $W_{1}(z)\equiv W_{2}(z).$ This result improves and generalizes the previous results given by Xuan and Gao.
Keywords:
value distribution theory, Nevanlinna theory, algebroid functions, uniqueness.
Received: 06.04.2017
Citation:
Ashok Rathod, “Nevanlinna's five-value theorem for algebroid functions”, Ufimsk. Mat. Zh., 10:2 (2018), 127–132; Ufa Math. J., 10:2 (2018), 127–132
Linking options:
https://www.mathnet.ru/eng/ufa429https://doi.org/10.13108/2018-10-2-127 https://www.mathnet.ru/eng/ufa/v10/i2/p127
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Abstract page: | 264 | Russian version PDF: | 87 | English version PDF: | 13 | References: | 24 |
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