Let $ f $ and $ g $ be two transcendental meromorphic functions of finite order with a Borel exceptional value $ \infty $, let $ \alpha $ $ (\not\equiv 0) $ be a small function of both $ f $ and $ g $, let $ d, k, n, m $ and $ v_j (j = 1, 2, \cdots, d) $ be positive integers, and let $ c_j (j = 1, 2, \cdots, d) $ be distinct nonzero finite values. If $ n\ge \max \{2k+m+\sigma+5, \sigma+2d+3\} $, where $ \sigma = v_1+v_2+\cdots +v_d $, and $ (f^n(z)(f^m(z)-1)\prod _{j = 1}^{d}f^{v_j}(z+c_j))^{(k)} $ and $ (g^n(z)(g^m(z)-1)\prod _{j = 1}^{d}g^{v_j}(z+c_j))^{(k)} $ share $ \alpha $ CM then $ f \equiv tg $, where $ t^m = t^{n+\sigma } = 1. $ This result extends and improves some restlts due to [
Citation: Zhiying He, Jianbin Xiao, Mingliang Fang. Unicity of transcendental meromorphic functions concerning differential-difference polynomials[J]. AIMS Mathematics, 2022, 7(5): 9232-9246. doi: 10.3934/math.2022511
Let $ f $ and $ g $ be two transcendental meromorphic functions of finite order with a Borel exceptional value $ \infty $, let $ \alpha $ $ (\not\equiv 0) $ be a small function of both $ f $ and $ g $, let $ d, k, n, m $ and $ v_j (j = 1, 2, \cdots, d) $ be positive integers, and let $ c_j (j = 1, 2, \cdots, d) $ be distinct nonzero finite values. If $ n\ge \max \{2k+m+\sigma+5, \sigma+2d+3\} $, where $ \sigma = v_1+v_2+\cdots +v_d $, and $ (f^n(z)(f^m(z)-1)\prod _{j = 1}^{d}f^{v_j}(z+c_j))^{(k)} $ and $ (g^n(z)(g^m(z)-1)\prod _{j = 1}^{d}g^{v_j}(z+c_j))^{(k)} $ share $ \alpha $ CM then $ f \equiv tg $, where $ t^m = t^{n+\sigma } = 1. $ This result extends and improves some restlts due to [
| [1] |
A. Banerjee, S. Majumder, On the uniqueness of certain types of differential-difference polynomials, Anal. Math., 43 (2017), 415–444. https://doi.org/10.1007/s10476-017-0402-3 doi: 10.1007/s10476-017-0402-3
|
| [2] | M. R. Chen, Z. X. Chen, Properties of difference polynomials of entire functions with finite order, Chinese Ann. Math. Ser. A, 33 (2012), 359–374. |
| [3] |
Y. M. Chiang, S. J. Feng, On the Nevanlinna characteristic of $f(z+\eta) $ and difference equations in the complex plane, Ramanujan J., 16 (2008), 105–129. https://doi.org/10.1007/s11139-007-9101-1 doi: 10.1007/s11139-007-9101-1
|
| [4] |
Y. M. Chiang, S. J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc., 361 (2009), 3767–3791. https://doi.org/10.1090/S0002-9947-09-04663-7 doi: 10.1090/S0002-9947-09-04663-7
|
| [5] |
M. L. Fang, H. Guo, On unique range sets for meromorphic or entire functions, Acta Math. Sin., 14 (1998), 569–576. https://doi.org/10.1007/BF02580416 doi: 10.1007/BF02580416
|
| [6] |
R. G. Halburd, R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487. https://doi.org/10.1016/j.jmaa.2005.04.010 doi: 10.1016/j.jmaa.2005.04.010
|
| [7] | R. G. Halburd, R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31 (2006), 463–478. |
| [8] |
R, G. Halburd, R. Korhonen, K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, T. Am. Math. Soc., 366 (2014), 4267–4298. https://doi.org/10.1090/S0002-9947-2014-05949-7 doi: 10.1090/S0002-9947-2014-05949-7
|
| [9] | W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. |
| [10] | V. Husna, S. Rajeshwari, S. M. Naveenkumar, A note on uniqueness of transcendental entire functions concerning differential-difference polynomials of finite order, Electron. J. Math. Anal. Appl., 9 (2021), 248–260. |
| [11] | I. Laine, Nevanlinna theory and complex differential equations, De Gruyter, Berlin, 1993. |
| [12] |
I. Laine, C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc., 76 (2007), 556–566. https://doi.org/10.1112/jlms/jdm073 doi: 10.1112/jlms/jdm073
|
| [13] |
Y. H. Li, J. Y. Qiao, The uniqueness of meromorphic functions concerning small functions, Sci. China Ser. A, 43 (2000), 581–590. https://doi.org/10.1007/BF02908769 doi: 10.1007/BF02908769
|
| [14] |
S. Majumder, S. Saha, A note on the uniqueness of certain types of differential-difference polynomials, Ukr. Math. J., 73 (2021), 679–694. https://doi.org/10.37863/umzh.v73i5.379 doi: 10.37863/umzh.v73i5.379
|
| [15] |
P. Sahoo, G. Biswas, Some results on uniqueness of entire functions concerning difference polynomials, Tamkang J. Math., 49 (2018), 85–97. https://doi.org/10.5556/j.tkjm.49.2018.2198 doi: 10.5556/j.tkjm.49.2018.2198
|
| [16] | C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers Group, Dordrecht, 2003. |
| [17] | L. Yang, Value distribution theory, Springer-Verlag, Berlin, 1993. |
| [18] |
J. L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl., 367 (2010), 401–408. https://doi.org/10.1016/j.jmaa.2010.01.038 doi: 10.1016/j.jmaa.2010.01.038
|
| [19] |
K. Y. Zhang, H. X. Yi, The value distribution and uniqueness of one certain type of differential-difference polynomials, Acta Math. Sci. Ser. B, 34 (2014), 719–728. https://doi.org/10.1016/S0252-9602(14)60043-6 doi: 10.1016/S0252-9602(14)60043-6
|
Zhiying He, Jianbin Xiao, Mingliang Fang. Unicity of transcendental meromorphic functions concerning differential-difference polynomials[J]. AIMS Mathematics, 2022, 7(5): 9232-9246. doi: 10.3934/math.2022511
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