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Ufimsk. Mat. Zh., 2018, Volume 10, Issue 2, Pages 127–132 (Mi ufa429)  

This article is cited in 2 scientific papers (total in 2 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*} (1+\frac{1}{k_{m}})\sum_{j=m}^{q}\frac{1}{1+k_{j}}+3\nu +\sum_{j=1}^{q}\delta_{j}<(q-m-1)(1+\frac{1}{k_{m}})+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.

Funding Agency Grant Number
University Grants Commission F1-17.1/2013-14-SC-KAR-40380
The author is supported by the UGC-Rajiv Gandhi National Fellowship (no. F1-17.1/2013-14-SC-KAR-40380) of India.


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English version:
Ufa Mathematical Journal, 2018, 10:2, 127–132 (PDF, 337 kB); https://doi.org/10.13108/2018-10-2-127

Bibliographic databases:

UDC: 512.5
MSC: 30D35
Received: 06.04.2017
Language:

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

Citation in format AMSBIB
\Bibitem{Rat18}
\by Ashok~Rathod
\paper Nevanlinna's five-value theorem for algebroid functions
\jour Ufimsk. Mat. Zh.
\yr 2018
\vol 10
\issue 2
\pages 127--132
\mathnet{http://mi.mathnet.ru/ufa429}
\transl
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 2
\pages 127--132
\crossref{https://doi.org/10.13108/2018-10-2-127}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000438890500010}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85048498813}


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    This publication is cited in the following articles:
    1. Ufa Math. J., 11:1 (2019), 121–132  mathnet  crossref  isi
    2. Ufa Math. J., 12:1 (2020), 114–120  mathnet  crossref  isi
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