Nevanlinna's five-value theorem for algebroid functions

Ashok Rathod

Ufa mathematical journal, Tome 10 (2018) no. 2, pp. 127-132

Résumé

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.

Détail

Keywords: value distribution theory, Nevanlinna theory, algebroid functions, uniqueness.

Bibliographie

[1] E. Ullrich, “Über den Einfluß der verzweigtheit einer algebloide auf ihre wertvertellung”, J. Reine Angew. Math., 1932:167 (1932), 198–220 | MR

[2] G. Valiron, “Sur quelques propriétés des fonctions algébroides”, Compt. Rend. Math., 189 (1929), 824–826 | Zbl

[3] N. Baganas, “Sur les valeurs algébriques dune fonctions algebroldes et les intégrales pseudo-abelinnes”, Annales Ecole Norm. Sup. Ser. 3, 66 (1949), 161–208 | DOI | MR | Zbl

[4] H.S. Gopalkrishna, S.S. Bhooanurmath, “Uniqueness theorems for meromorphic functions”, Math. Scand., 39 (1976), 125–130 | DOI | MR

[5] Lahiri Indrajit, Pal Rupa, “A note on Nevanlinna's five value theorerm”, Bull. Korean Math. Soc., 52:2 (2015), 345–350 | DOI | MR | Zbl

[6] Yu-Zan He, Ye-Zhou Li, “Some results on algebroid functions”, Comp. Variab. Ellip. Equat., 43:3-4 (2001), 299–313 | MR

[7] S. Daochun, G. Zongsheng, “On the operation of algebroid functions”, Acta Math. Sci., 30:1 (2010), 247–256 | DOI | MR | Zbl

[8] S. Daochun, G. Zongsheng, Value disribution theory of algebroid functions, Science Press, Beijing, 2014

[9] Yu-Zan He, Xiu-Zhi Xiao, Algebroid functions and Ordinarry Difference Equations, Science Press, Beijing, 1988

[10] S. Daochun, G. Zongsheng, “Theorems for algebroid functions”, Acta Math. Sinica, 49:5 (2006), 1–6 | MR

[11] Y. Hongxun, “On the multiple values and uniqueness of algebroid functions”, Chinese J. Eng. Math., 8 (1991), 1–8

[12] W.K. Hayman, Meromorphic functions, Oxford University Press, Oxford, 1964 | MR | Zbl

[13] F. Minglang, “Unicity theorem for algebroid functions”, Acta. Math. Sinica, 3:6 (1993), 217–222

[14] Pingyuan Zhang, Peichu Hu, “On uniqueness for algebroid functions of finite order”, Acta. Math. Sinica, 35:3 (2015), 630–638 | MR | Zbl

[15] G.S. Prokoporich, “Fix-points of meromorphic or entire functions”, Ukrainian Math. J., 25:2 (1973), 248–260 | MR

[16] Z. Qingcai, “Uniqueness of algebroid functions”, Math. Pract. Theory, 43:1 (2003), 183–187

[17] Cao Tingbin, Yi Hongxun, “On the uniqueness theory of algebroid functions”, Southest Asian Bull. Math., 33:1 (2009), 25–39 | MR | Zbl

[18] Zu-Xing Xuan, ZongG-Sheng Gao, “Uniqueness theorems for algebroid functions”, Compl. Variab. Ellipt. Equat., 51:7 (2006), 701–712 | DOI | MR | Zbl

[19] C.C. Yang, H.X. Yi, Uniqueness theory of meromorphic functions, Math. Appl., 557, Kluwer Academic Publishers, Dordrecht, 2003 | MR | Zbl

[20] H.X. Yi, “The multiple values of meromorphic functions and uniqueness”, Chinese Ann. Math. Ser. A, 10:4 (1989), 421–427 | MR | Zbl

[21] R.S. Dyavanal, Ashok Rathod, “Some generalisation of Nevanlinna's five-value theorem algebroid functions on annuli”, Asian J. Math. Comp. Resear., 20:2 (2017), 85–95

[22] R.S. Dyavanal, Ashok Rathod, “Nevanlinna's five-value heorem for derivatives of meromorphic functions sharing values on annuli”, Asian J. Math. Comp. Resear., 20:1 (2017), 13–21

[23] R.S. Dyavanal, Ashok Rathod, “Unicity theorem for algebroid functions related to multiple values and derivatives on annuli”, Int. J. Fuzzy Math. Arch., 13:1 (2017), 25–39 | MR

[24] Ashok Rathod, “Several uniqueness theorems for algebroid functions”, J. Anal., 25:2 (2017), 203–213 | DOI | MR | Zbl

[25] Ashok Rathod, “The multiple values of algebroid functions and uniqueness”, Asian J. Math. Comp. Resear., 14:2 (2016), 150–157 | MR

[26] Ashok Rathod, “The multiple values of algebroid functions and uniqueness on annuli”, Konoralf J. Math., 5:2 (2017), 216–227 | MR | Zbl

[27] Ashok Rathod, “On the deficiencies of algebriod functions and their differential polynomials”, J. Basic Appl. Resear. Int., 1:1 (2016), 1–11