Title:
|
Generalizations on the results of Cao and Zhang (English) |
Author:
|
Majumder, Sujoy |
Author:
|
Mandal, Rajib |
Language:
|
English |
Journal:
|
Mathematica Bohemica |
ISSN:
|
0862-7959 (print) |
ISSN:
|
2464-7136 (online) |
Volume:
|
147 |
Issue:
|
1 |
Year:
|
2022 |
Pages:
|
65-94 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod _{i=1}^{k}(f^{(i)})^{n_{i}}$ and $P(g)\prod _{i=1}^{k}(g^{(i)})^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp. (English) |
Keyword:
|
meromorphic function |
Keyword:
|
uniqueness |
Keyword:
|
weighted sharing |
Keyword:
|
differential polynomial |
MSC:
|
30D35 |
idZBL:
|
Zbl 07547242 |
idMR:
|
MR4387469 |
DOI:
|
10.21136/MB.2021.0072-19 |
. |
Date available:
|
2022-04-17T18:09:37Z |
Last updated:
|
2022-09-06 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149592 |
. |
Reference:
|
[1] Alzahary, T. C., Yi, H. X.: Weighted value sharing and a question of I. Lahiri.Complex Variables, Theory Appl. 49 (2004), 1063-1078. Zbl 1067.30055, MR 2111304, 10.1080/02781070410001701074 |
Reference:
|
[2] Banerjee, A.: On a question of Gross.J. Math. Anal. Appl. 327 (2007), 1273-1283. Zbl 1115.30029, MR 2280003, 10.1016/j.jmaa.2006.04.078 |
Reference:
|
[3] Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order.Rev. Mat. Iberoam. 11 (1995), 355-373. Zbl 0830.30016, MR 1344897, 10.4171/RMI/176 |
Reference:
|
[4] Cao, Y.-H., Zhang, X.-B.: Uniqueness of meromorphic functions sharing two values.J. Inequal. Appl. 2012 (2012), Article ID 100, 10 pages. Zbl 1291.30193, MR 2946490, 10.1186/1029-242X-2012-100 |
Reference:
|
[5] Chen, H., Fang, M.: The value distribution of $f^{n}f'$.Sci. China, Ser. A. 38 (1995), 789-798 \99999ZBL99999 0839. 30026. MR 1360682 |
Reference:
|
[6] Fang, M., Hua, X.: Entire functions that share one value.J. Nanjing Univ., Math. Biq. 13 (1996), 44-48. Zbl 0899.30022, MR 1411806 |
Reference:
|
[7] Fang, M., Qiu, H.: Meromorphic functions that share fixed-points.J. Math. Anal. Appl. 268 (2002), 426-439. Zbl 1030.30028, MR 1896207, 10.1006/jmaa.2000.7270 |
Reference:
|
[8] Hayman, W. K.: Meromorphic Functions.Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). Zbl 0115.06203, MR 0164038 |
Reference:
|
[9] Köhler, I.: Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros.Complex Variables, Theory Appl. 11 (1989), 39-48. Zbl 0637.30029, MR 0998240, 10.1080/17476938908814322 |
Reference:
|
[10] Lahiri, I.: Weighted value sharing and uniqueness of meromorphic functions.Complex Variables, Theory Appl. 46 (2001), 241-253. Zbl 1025.30027, MR 1869738, 10.1080/17476930108815411 |
Reference:
|
[11] Lahiri, I., Dewan, S.: Value distribution of the product of a meromorphic function and its derivative.Kodai Math. J. 26 (2003), 95-100. Zbl 1077.30025, MR 1966685, 10.2996/kmj/1050496651 |
Reference:
|
[12] Lahiri, I., Sarkar, A.: Nonlinear differential polynomials sharing 1-points with weight two.Chin. J. Contemp. Math. 25 (2004), 325-334. Zbl 1069.30051, MR 2159311 |
Reference:
|
[13] Xu, J., Yi, H., Zhang, Z.: Some inequalities of differential polynomials.Math. Inequal. Appl. 12 (2009), 99-113. Zbl 1169.30314, MR 2489354, 10.7153/mia-12-09 |
Reference:
|
[14] Yamanoi, K.: The second main theorem for small functions and related problems.Acta Math. 192 (2004), 225-294. Zbl 1203.30035, MR 2096455, 10.1007/BF02392741 |
Reference:
|
[15] Yang, C.-C.: On deficiencies of differential polynomials II.Math. Z. 125 (1972), 107-112. Zbl 0217.38402, MR 294642, 10.1007/BF01110921 |
Reference:
|
[16] Yang, C.-C., Hua, X.: Uniqueness and value-sharing of meromorphic functions.Ann. Acad. Sci. Fenn., Math. 22 (1997), 395-406. Zbl 0890.30019, MR 1469799 |
Reference:
|
[17] Yang, C.-C., Yi, H.-X.: Uniqueness Theory of Meromorphic Functions.Mathematics and its Applications 557. Kluwer Academic Publishers, Dordrecht (2003). Zbl 1070.30011, MR 2105668, 10.1007/978-94-017-3626-8 |
Reference:
|
[18] Yi, H.-X.: On characteristic function of a meromorphic function and its derivative.Indian J. Math. 33 (1991), 119-133. Zbl 0799.30018, MR 1140875 |
Reference:
|
[19] Zhang, Q.: Meromorphic function that shares one small function with its derivative.JIPAN, J. Inequal. Pure Appl. Math. 6 (2005), Article ID 116, 13 pages. Zbl 1097.30033, MR 2178297 |
Reference:
|
[20] Zhang, Z., Li, W.: Picard exceptional values for two differential polynomials.Acta Math. Sin. 37 (1994), 828-835 Chinese. Zbl 0815.30021, MR 1339161 |
. |