Suppose that $ f(z) $ is a meromorphic function with hyper order $ \sigma_{2}(f) < 1 $. Let $ L(z, f) = b_1(z)f(z+c_1)+b_2(z)f(z+c_2)+\cdots+b_n(z)f(z+c_n) $ be a linear difference polynomial, where $ b_1(z), b_2(z), \cdots, b_n(z) $ are nonzero small functions relative to $ f(z) $, and $ c_1, c_2, \cdots, c_n $ are distinct complex numbers. We investigate the uniqueness results about $ f(z) $ and $ L(z, f) $ sharing small functions. These results promote the existing results on differential cases and difference cases of Brück conjecture. Some sufficient conditions to show that $ f(z) $ and $ L(z, f) $ cannot share some small functions are also presented.
Citation: Ran Ran Zhang, Chuang Xin Chen, Zhi Bo Huang. Uniqueness on linear difference polynomials of meromorphic functions[J]. AIMS Mathematics, 2021, 6(4): 3874-3888. doi: 10.3934/math.2021230
Suppose that $ f(z) $ is a meromorphic function with hyper order $ \sigma_{2}(f) < 1 $. Let $ L(z, f) = b_1(z)f(z+c_1)+b_2(z)f(z+c_2)+\cdots+b_n(z)f(z+c_n) $ be a linear difference polynomial, where $ b_1(z), b_2(z), \cdots, b_n(z) $ are nonzero small functions relative to $ f(z) $, and $ c_1, c_2, \cdots, c_n $ are distinct complex numbers. We investigate the uniqueness results about $ f(z) $ and $ L(z, f) $ sharing small functions. These results promote the existing results on differential cases and difference cases of Brück conjecture. Some sufficient conditions to show that $ f(z) $ and $ L(z, f) $ cannot share some small functions are also presented.
| [1] | M. Ablowitz, R. G. Halburd, B. Herbst, On the extension of Painlev$\acute{ e }$ property to difference equations, Nonlinearity, 13 (2000), 889–905. |
| [2] | W. Bergweiler, J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge, 142 (2007), 133–147. |
| [3] | R. Brück, On entire functions which share one value CM with their first derivate, Results Math., 30 (1996), 21–24. |
| [4] | K. S. Charak, R. J. Korhonen, G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, J. Math. Anal. Appl., 435 (2016), 1241–1248. |
| [5] | Z. X. Chen, K. H. Shon, On conjecture of R. Brück concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math., 8 (2004), 235–244. |
| [6] | Z. X. Chen, On the difference counterpart of Brück's conjecture, Acta Math. Sci., 34 (2014), 653–659. |
| [7] |
N. Cui, Z. X. Chen, The conjecture on unity of meromorphic functions concerning their differences, J. Differ. Equ. Appl., 22 (2016), 1452–1471. doi: 10.1080/10236198.2016.1201477
|
| [8] | A. Edrei, W. H. J. Fuchs, On the growth of meromorphic functions with several deficient values, T. Am. Math. Soc., 93 (1959), 292–328. |
| [9] | A. A. Gol'dberg, I. V. Ostrovskii, The distribution of values of meromorphic functions, Moscow: Nauka, 1970. |
| [10] | F. Gross, Factorization of meromorphic functions, Washington: U. S. Government Printing Office, 1972. |
| [11] |
G. Gundersen, Meromorphic functions that share four values, T. Am. Math. Soc., 277 (1983), 545–567. doi: 10.1090/S0002-9947-1983-0694375-0
|
| [12] |
G. Gundersen, L. Z. Yang, Entire functions that share one values with one or two of their derivatives, J. Math. Anal. Appl., 223 (1998), 88–95. doi: 10.1006/jmaa.1998.5959
|
| [13] |
R. G. Halburd, R. J. Korhonen, K. Tohge, Holomorphic curves with shift-invariant hyper-plane preimages, T. Am. Math. Soc., 366 (2014), 4267–4298. doi: 10.1090/S0002-9947-2014-05949-7
|
| [14] | W. K. Hayman, Meromorphic functions, Oxford: Clarendon Press, 1964. |
| [15] |
J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. L. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient condition for periodicity, J. Math. Anal. Appl., 355 (2009), 352–363. doi: 10.1016/j.jmaa.2009.01.053
|
| [16] |
J. Heittokangas, R. K. Korhonen, I. Laine, J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic, 56 (2011), 81–92. doi: 10.1080/17476930903394770
|
| [17] |
Z. B. Huang, R. R. Zhang, Uniqueness of the differences of meromorphic functions, Anal. Math., 44 (2018), 461–473. doi: 10.1007/s10476-018-0306-x
|
| [18] |
P. Li, W. J. Wang, Entire functions that share a small function with its derivative, J. Math. Anal. Appl., 328 (2007), 743–751. doi: 10.1016/j.jmaa.2006.04.083
|
| [19] |
X. M. Li, H. X. Yi, C. Y. Kang, Notes on entire functions sharing an entire function of a smaller order with their difference operators, Arch. Math., 99 (2012), 261–270. doi: 10.1007/s00013-012-0425-8
|
| [20] | E. Mues, Meromorphic functions sharing four values, Complex Var. Elliptic, 12 (1989), 167–179. |
| [21] |
R. Nevanlinna, Einige Eindeutigkeitssätze in der theorie der meromorphen funktionen, Acta Math., 48 (1926), 367–391. doi: 10.1007/BF02565342
|
| [22] | L. A. Rubel, C. C. Yang, Value shared by an entire function and its derivative, Berlin: Springer, 1977. |
| [23] |
R. Ullah, X. M. Li, F. Faizullah, H. X. Yi, R. A. Khan, On the uniqueness results and value distribution of meromorphic mappings, Mathematics, 5 (2017), 42. doi: 10.3390/math5030042
|
| [24] |
S. Wang, Meromorphic functions sharing four values, J. Math. Anal. Appl., 173 (1993), 359–369. doi: 10.1006/jmaa.1993.1072
|
| [25] | C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Dordrecht: Kluwer Academic Publishers Group, 2003. |
| [26] |
L. Z. Yang, Entire functions that share finite values with their derivatives, Bull. Aust. Math. Soc., 41 (1990), 337–342. doi: 10.1017/S0004972700018190
|
| [27] |
L. Z. Yang, J. L. Zhang, Non-existence of meromorphic solution of a Fermat type functional equation, Aequationes Math., 76 (2008), 140–150. doi: 10.1007/s00010-007-2913-7
|
| [28] | J. Zhang, H. Y. Kang, L. W. Liao, Entire functions sharing a small entire function with their difference operators. Bull. Iran. Math. Soc., 41 (2015), 1121–1129. |
Ran Ran Zhang, Chuang Xin Chen, Zhi Bo Huang. Uniqueness on linear difference polynomials of meromorphic functions[J]. AIMS Mathematics, 2021, 6(4): 3874-3888. doi: 10.3934/math.2021230
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