Characteristic function and deficiency of algebroid functions on annuli

Ashok Rathod

Ufa mathematical journal, Tome 11 (2019) no. 1, pp. 121-132

Résumé

In this paper, we develop the value distribution theory for meromorphic functions with maximal deficiency sum for algebroid functions on annuli and we study the relationship between the deficiency of algebroid function on annuli and that of their derivatives. Let $W(z)$ be an $\nu$-valued algebroid function on the annulus $\mathbb{A}\left(\frac{1}{R_{0}},R_{0}\right)$ $(1$ with maximal deficiency sum and the order of $W(z)$ is finite. Then i. $\limsup\limits_{r\rightarrow\infty}\frac{T_{0}(r,W')}{T_{0}(r,W)}= 2-\delta_{0}(\infty,W)-\theta_{0}(\infty,W);$ ii. $\limsup\limits_{r\rightarrow\infty}\frac{N_{0}(r,\frac{1}{W'})}{T_{0}(r,W')}=0;$ iii. $\frac{1-\delta_{0}(\infty,W)}{2-\delta_{0}(\infty,W)}\leq K_{0}(W')\leq \frac{2(1-\delta_{0}(\infty,W))}{2-\delta_{0}(\infty,W)},$ where $$K_{0}(W')=\limsup\limits_{r\rightarrow\infty}\frac{N_{0}(r,W')+N_{0}(r,\frac{1}{W'})}{T_{0}(r,W')}.$$

Détail

Keywords: Nevanlinna Theory, maximal deficiency sum, algebroid functions, the annuli.

Bibliographie

[1] E. Ullrich, “Über den Einflußder 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 d'une fonctions algebroldes et les intégrales pseudo-abelinnes”, Annales École Norm. Sup. Sér. III, 66 (1949), 161–208 | DOI | MR | Zbl

[4] M. L. Fang, “A note on a result of Singh and Kulkarni”, Int. J. Math. Sci., 23:4 (2000), 285–288 | DOI | MR | Zbl

[5] S.K. Singh, V.N. Kulkarni, “Characteristic function of a meromorphic function and its derivative”, Ann Polon Math., 28 (1973), 123–133 | DOI | MR | Zbl

[6] Yu-Zan He, Ye-Zhou Li, Compl. Variab. Theory Appl., 43:3–4 (2001), Some results on algebroid functions | 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] Meili Liang, “On the value distribution of algebroid functions”, SOP Trans. Appl. Math., 1:1 (2014)

[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, Zong G-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] Zhaojun Wu, Sheng'an Chen, “Characteristic function and deficiency of meromorphic functions in the punctured plane”, Acta Math. Scientia, 35:B(3) (2015), 673–680 | DOI | MR | Zbl

[21] R. S. Dyavanal, Ashok Rathod, “Uniqueness theorems for meromorphic functions on annuli”, Indian J. Math. Math. Sci., 12:1 (2016), 1–10 | MR

[22] R. S. Dyavanal, Ashok Rathod, “Multiple values and uniqueness of meromorphic functions on annuli”, Int. J. Pure Appl. Math., 107:4 (2016), 983–995 | DOI

[23] R. S. Dyavanal, Ashok Rathod, “On the value distribution of meromorphic functions on annuli”, Indian J. Math. Math. Sci., 12:2 (2016), 203–217 | MR

[24] 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 | MR

[25] 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 | MR

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

[27] R. S. Dyavanal, Ashok Rathod, “General Milloux inequality for algebroid functions on annuli”, Int. J. Math. Appl., 5:3 (2017), 319–326 | MR

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

[29] Ashok Rathod, “The uniqueness of meromorphic functions concerning five or more values and small functions on annuli”, Asian J. Current Res., 1:3 (2016), 101–107

[30] Ashok Rathod, “Uniqueness of algebroid functions dealing with multiple values on annuli”, J. Basic Appl. Res. Int., 19:3 (2016), 157–167

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

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

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

[34] Ashok Rathod, “Nevanlinna's five-value theorem for algebroid functions”, Ufa Math. J., 10:2 (2018), 127–132 | DOI | MR

[35] Ashok Rathod, “Nevanlinna's five-value theorem for derivatives of algebroid functions on annuli”, Tamkang J. Mathe., 49:2 (2018), 129–142 | DOI | MR | Zbl

[36] S. S. Bhoosnurmath, R. S. Dyavanal, Mahesh Barki, Ashok Rathod, “Value distribution for n-th difference operator of meromorphic functions with maximal deficiency sum”, J. Anal., 2018, 1–15