On the Study of Meromorphic Functions That Shares Small Functions Partially With the Second Order Difference Operator
DOI:
https://doi.org/10.26713/cma.v13i3.1913Keywords:
Uniqueness, Meromorphic function, Partial sharing, Small function, Difference operatorAbstract
In this paper, we looked at some problems with the uniqueness of meromorphic functions with a second order difference operator. We looked at them from the point of view of partial sharing. We have obtained two uniqueness results. In the first theorem \(\Delta^2 \mathfrak{g(z)}\) and \(\mathfrak{g(z)}\) shares \(\mathfrak{a}_1\mathfrak{(z)}\), \(\mathfrak{a}_2\mathfrak{(z)}\), \(\infty\) CM, whereas in the second theorem \(\mathfrak{g(z)}\) and \(\Delta^2 \mathfrak{g(z)}\) partially share \(\mathfrak{a}_1\mathfrak{(z)}\), \(\mathfrak{a}_2\mathfrak{(z)}\) CM that generalizes the results due to Banerjee and Maity (Meromorphic function partially shares small functions or values with its linear c-shift operator, Bulletin of the Korean Mathematical Society 58(5) (2021), 1175 -- 1192), and Heittokangas et al., Uniqueness of meromorphic functions sharing values with their shifts, Complex Variables and Elliptic Equations 56(1-4) (2011), 81 -- 92.
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A. Banerjee and S. Maity, Meromorphic function partially shares small functions or values with its linear c-shift operator, Bulletin of the Korean Mathematical Society 58(5) (2021), 1175 – 1192, DOI: 10.4134/BKMS.b200840.
K. S. Charak, R. J. Korhonen and G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, Journal of Mathematical Analysis and Applications 435(2) (2016), 1241 – 1248, DOI: 10.1016/j.jmaa.2015.10.069.
Z. X. Chen and H. X. Yi, On sharing values of meromorphic functions and their differences, Results in Mathematics 63(1) (2013), 557 – 565, DOI: 10.1007/s00025-011-0217-7.
Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f (z +η) and difference equations in the complex plane, The Ramanujan Journal 16(1) (2008), 105 – 29, DOI: 10.1007/s11139-007-9101-1.
G. G. Gundersen, Meromorphic functions that share three values IM and a fourth value CM, Complex Variables, Theory and Application: An International Journal 20(1-4) (1992), 99 – 106, DOI: 10.1080/17476939208814590.
R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, Journal of Mathematical Analysis and Applications 314(2) (2006), 477 – 87, DOI: 10.1016/j.jmaa.2005.04.010.
W. K. Hayman and E. F. Lingham, Meromorphic functions, in: Research Problems in Function Theory. Problem Books in Mathematics, pp. 1 – 22, Springer, Cham., (2019), DOI: 10.1007/978-3-030-25165-9_1.
J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Variables and Elliptic Equations: An International Journal 56(1-4) (2011), 81 – 92, DOI: 10.1080/17476930903394770.
J. Heittokangas, R. Korhonen, I. Laine,J. Rieppo, and J. Zhang, Value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity, Journal of Mathematical Analysis and Applications 355(1) (2009), 352 – 363, DOI: 10.1016/j.jmaa.2009.01.053.
Z.-B. Huang and R.-R. Zhang, Uniqueness of the differences of meromorphic functions, Analysis Mathematica 44(4) (2018), 461 – 473, DOI: 10.1007/s10476-018-0306-x.
Y. Jiang and Z. Chen, Meromorphic functions share two values with its difference operator, Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica 63(1) (2017), 169 – 175, DOI: 10.2478/aicu-2014-0056.
I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics series, Vol. 15 De Gruyter, Berlin—New York (1993), DOI: 10.1515/9783110863147.
K. Liu, Meromorphic functions sharing a set with applications to difference equations, Journal of Mathematical Analysis and Applications 359(1) (2009), 384 – 393, DOI: 10.1016/j.jmaa.2009.05.061.
F. Lü and W. Lü, Meromorphic functions sharing three values with their difference operators, Computational Methods and Function Theory 17(3) (2017), 395 – 403, DOI: 10.1007/s40315-016-0188-5.
V. Noulorvang and D. T. Pham, On partial value sharing results of meromorphic functions with their shifts and its applications, Bulletin of the Korean Mathematical Society 57(5) (2020), 1083 – 1094, DOI: 10.4134/BKMS.b190483.
K. Yamanoi, The second main theorem for small functions and related problems, Acta Mathematica 192(2) (2004), 225 – 294, DOI: 10.1007/BF02392741.
C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions, Springer Science and Business Media, Springer, 557 (2003), DOI: 10.1007/978-94-017-3626-8.
J. Zhang and L. Liao, Entire functions sharing some values with their difference operators, Science China Mathematics 57(10) (2014), 2143 – 2152, DOI: 10.1007/s11425-014-4848-5.
J. Zhang and R. Korhonen, On the Nevanlinna characteristic of f (qz) and its applications, Journal of Mathematical Analysis and Applications 369(2) (2010), 537 – 544, DOI: 10.1016/j.jmaa.2010.03.038.
J. Zhang, Value distribution and shared sets of differences of meromorphic functions, Journal of Mathematical Analysis and Applications 367(2) (2010), 401 – 408, DOI: 10.1016/j.jmaa.2010.01.038.
L. Zhen, Meromorphic Functions sharing three polynomials with their difference operators, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 54(5) (2019), 296 – 301, DOI: 10.3103/S1068362319050066.
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