Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Growth of Transcendental Meromorphic Solutions of Certain algebraic Difference Equations

  • Xinjun Yao (Department of Mathematics, Shaoxing College of Arts and Sciences) ;
  • Yong Liu (Department of Mathematics, Shaoxing College of Arts and Sciences) ;
  • Chaofeng Gao (Department of Mathematics, Shaoxing College of Arts and Sciences)
  • Received : 2023.03.08
  • Accepted : 2023.06.27
  • Published : 2024.03.31
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Abstract

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), bi(z) for i = 0, 1, 2 and dj (z) for j = 0, ..., 4 are given functions, △cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the bi(z) and the dj(z) are polynomials, and d4(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d0(z) + 1, or, deg a(z) = deg d0(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

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Acknowledgement

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

References

  1. J. Clunie, On integral and meromorphic functions, J. Lond. Math. Soc., 37(1962), 17-27.  https://doi.org/10.1112/jlms/s1-37.1.17
  2. Y. M. Chiang and S. J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc., 361(7)(2009), 3767-3791.  https://doi.org/10.1090/S0002-9947-09-04663-7
  3. X. Dong and K. Liu, Entire function sharing a small function with its mixed-operators, Georgian Math. J., 26(2019), 47-62.  https://doi.org/10.1515/gmj-2017-0024
  4. X. M. Gui, H. Y. Xu and H. Wang, uniqueness of meromorphic functions sharing small functions in the k-punctured complex plane, AIMS Mathematics, 5(6)(2020), 7438-7457.  https://doi.org/10.3934/math.2020476
  5. R. G. Halburd and R. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31(2006), 463-478. 
  6. R. G. Halburd and R. J. Korhonen, Finite order solutions and the discrete Painleve equations, Proc. London Math. Soc., 94(2)(2007), 443-474.  https://doi.org/10.1112/plms/pdl012
  7. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. 
  8. K. Ishizaki and N. Yanagihara, Wiman-Valiron method for difference equations, Nagoya Math. J., 175(2004), 75-102.  https://doi.org/10.1017/S0027763000008916
  9. I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc., 76(2)(2007), 556-566.  https://doi.org/10.1112/jlms/jdm073
  10. X. M. Li and H. X. Yi, Entire functions sharing an entire function of smaller order with their difference operators, Acta Mathematical Sinica, English Series., 30(3)(2014), 481-498.  https://doi.org/10.1007/s10114-014-2042-x
  11. L. W. Liao and C. C. Yang, On the growth and factorization of entire solutions of algebraic differential equations, Ann. Acad. Sci. Fenn. Math., 25(2000), 73-84. 
  12. A. Z. Mohono and V. D. Mohono, Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations, Sibirsk. Mat. Zh., 15(1974) 1305-22. 
  13. J. Malmquist, Sur les fonctions a un nombre fini des branches definies par les equations differenielles du premier ordre, Acta Math. 36(1913), 297-343.  https://doi.org/10.1007/BF02422385
  14. N. Steinmetz, Ein Malmquistscher Satz fur algebraische Differentialgleichungen erster Ordnung, J. Reine Angew. Math., 316(1980), 44-53. 
  15. H. Wittich, Einige Eigenschaften der Losungen von w = a(z) +b(z)w +c(z)w2, Arch. Math., 5(1954), 226-232.  https://doi.org/10.1007/BF01899342
  16. C. C. Yang and H. X. Yi, Uniqueness of Meromorphic Functions, Kluwer, Dordrecht, 2003.