| Title:
|
Uniqueness of meromorphic functions concerning value sharing of nonlinear differential monomials (English) |
| Author:
|
Majumder, Sujoy |
| Author:
|
Mandal, Rajib |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
281-304 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^{n}(f^{(k)})^{m}-p$ and $g^{n}(g^{(k)})^{m}-p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018). (English) |
| Keyword:
|
uniqueness |
| Keyword:
|
meromorphic function |
| Keyword:
|
small function |
| Keyword:
|
nonlinear differential polynomial |
| Keyword:
|
normal family |
| MSC:
|
30D30 |
| MSC:
|
30D35 |
| idZBL:
|
07250711 |
| idMR:
|
MR4221835 |
| DOI:
|
10.21136/MB.2019.0010-18 |
| . |
| Date available:
|
2020-09-14T15:02:00Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148350 |
| . |
| 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] Banerjee, A., Majumder, S.: On certain non-linear differential polynomial sharing a non-zero polynomial.Bol. Soc. Mat. Mex., III. Ser. 24 (2018), 155-180. Zbl 1390.30036, MR 3773104, 10.1007/s40590-016-0156-0 |
| Reference:
|
[4] 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:
|
[5] Cao, Y.-H., Zhang, X.-B.: Uniqueness of meromorphic functions sharing two values.J. Inequal. Appl. 2012 (2012), Paper No. 100, 10 pages. Zbl 1291.30193, MR 2946490, 10.1186/1029-242X-2012-100 |
| Reference:
|
[6] Chang, J., Zalcman, L.: Meromorphic functions that share a set with their derivatives.J. Math. Anal. Appl. 338 (2008), 1020-1028. Zbl 1160.30014, MR 2386477, 10.1016/j.jmaa.2007.05.079 |
| Reference:
|
[7] Chen, H., Fang, M.: The value distribution of $f\sp n f'$.Sci. China, Ser. A 38 (1995), 789-798. Zbl 0839.30026, MR 1360682 |
| Reference:
|
[8] Dou, J., Qi, X.-G., Yang, L.-Z.: Entire functions that share fixed-points.Bull. Malays. Math. Sci. Soc. (2) 34 (2011), 355-367. Zbl 1216.30033, MR 2788409 |
| Reference:
|
[9] 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:
|
[10] 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:
|
[11] Hayman, W. K.: Meromorphic Functions.Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). Zbl 0115.06203, MR 0164038 |
| Reference:
|
[12] Köhler, L.: 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:
|
[13] Lahiri, I.: Weighted sharing and uniqueness of meromorphic functions.Nagoya Math. J. 161 (2001), 193-206. Zbl 0981.30023, MR 1820218, 10.1017/S0027763000027215 |
| Reference:
|
[14] 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:
|
[15] Lahiri, I., Dewan, S.: Inequalities arising out of the value distribution of a differential monomial.JIPAM, J. Inequal. Pure Appl. Math. 4 (2003), Paper No. 27, 6 pages. Zbl 1126.30309, MR 1994240 |
| Reference:
|
[16] 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:
|
[17] 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:
|
[18] Lin, W., Yi, H. X.: Uniqueness theorems for meromorphic functions concerning fixed-point.Complex Variables, Theory Appl. 49 (2004), 793-806. Zbl 1067.30065, MR 2097218, 10.1080/02781070412331298624 |
| Reference:
|
[19] Pang, X. C.: Normality conditions for differential polynomials.Kexue Tongbao, Sci. Bull. 33 (1988), 1690-1693. Zbl 1382.30060, MR 0999887 |
| Reference:
|
[20] Schiff, J. L.: Normal Families.Universitext. Springer, New York (1993). Zbl 0770.30002, MR 1211641, 10.1007/978-1-4612-0907-2 |
| Reference:
|
[21] 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:
|
[22] 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:
|
[23] Yang, C. C.: On deficiencies of differential polynomials. II.Math. Z. 125 (1972), 107-112. Zbl 0217.38402, MR 0294642, 10.1007/BF01110921 |
| Reference:
|
[24] 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:
|
[25] 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:
|
[26] 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:
|
[27] Zalcman, L.: A heuristic principle in complex function theory.Am. Math. Mon. 82 (1975), 813-817. Zbl 0315.30036, MR 0379852, 10.2307/2319796 |
| Reference:
|
[28] Zhang, J.: Uniqueness theorems for entire functions concerning fixed points.Comput. Math. Appl. 56 (2008), 3079-3087. Zbl 1165.30362, MR 2474563, 10.1016/j.camwa.2008.07.006 |
| Reference:
|
[29] Zhang, Q.: Meromorphic function that shares one small function with its derivative.JIPAM, J. Inequal. Pure Appl. Math. 6 (2005), Article No. 116, 13 pages. Zbl 1097.30033, MR 2178297 |
| Reference:
|
[30] Zhang, T., Lü, W.: Uniqueness theorems on meromorphic functions sharing one value.Comput. Math. Appl. 55 (2008), 2981-2992. Zbl 1142.30330, MR 2401445, 10.1016/j.camwa.2007.11.029 |
| Reference:
|
[31] Zhang, X.-Y., Lin, W.-C.: Corrigendum to ``Uniqueness and value-sharing of entire functions''.J. Math. Anal. Appl. 352 (2009), page 971. Zbl 1160.30345, MR 2501943, 10.1016/j.jmaa.2008.10.051 |
| . |
