Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ORDER, TYPE AND ZEROS OF ANALYTIC AND MEROMORPHIC FUNCTIONS OF [p, q] - ϕ ORDER IN THE UNIT DISC

  • Received : 2022.11.10
  • Accepted : 2023.06.21
  • Published : 2023.06.30
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Abstract

In this paper, we investigate the [p, q] - φ order and [p, q] - φ type of f1 + f1, ${\frac{f_1}{f_2}}$ and f1 f1, where f1 and f1 are analytic or meromorphic functions with the same [p, q]-φ order and different [p, q]-φ type in the unit disc. Also, we study the [p, q]-φ order and [p, q]-φ type of different f and its derivative. At the end, we investigate the relationship between two different [p, q] - φ convergence exponents of f. We extend some earlier precedent well known results.

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Acknowledgement

The authors are very much thankful to the referees for their valuable comments towards the improvement of the paper.

References

  1. S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential euqations, Composition Math. 25 (1972), 61-70.
  2. B. Belaidi, Growth of solutions to linear equations with analytic coefficients of [p,q]-order in the unit disc, Electron. J. Diff. Equ. 156 (2) (2011), 25-38.
  3. T.B. Cao, and H.X. Yi, The growth of solutions of linear equations with coefficients of iterated order in the unit disc, J. Math. Anl. Appl. 319 (2006), 278-294. https://doi.org/10.1016/j.jmaa.2005.09.050
  4. I. Chyzhykov, J. Heittokangas, J. Rattya, Finiteness of φ- order of solutions of linear differential equations in the unit disc, J.Anal.Math. 109 (2009), 163-198. https://doi.org/10.1007/s11854-009-0030-3
  5. W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  6. W. Hayman, On the characteristic of functions meromorphic in the unit disc and of their integrals, Acta Math., 112 (1964), 181-214. https://doi.org/10.1007/BF02391770
  7. J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1-54.
  8. J. Heittokangas,R. Korhonen. and J. Rattya, Fast growing solutions of linear differential equations in the unit disc, Results Math., 49 (2006), 265-278. https://doi.org/10.1007/s00025-006-0223-3
  9. O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math., 282 (1976), 53-67.
  10. O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-type and lower (p,q)-type of an entire function, J. Reine Angew. Math., 290 (1977), 180-190.
  11. I. Lain, Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, 1993.
  12. B.Ja. Levin, Distribution of Zeros of Entire Functions, revised edition Transl. Math. Monographs, Vol. 5, Amer. Math. Soc. Providence, 1980.
  13. L.M. Li and T.B. Cao, Solutions for differential equations with moromorphic coefficients of (p,q)-order in the plane, Electron. J. Diff. Equ., 2012 (195) (2012), 1-15. https://doi.org/10.1186/1687-1847-2012-1
  14. C. Linden, The representation of regular functions, J.London Math. Soc., 39 (1964), 19-30. https://doi.org/10.1112/jlms/s1-39.1.19
  15. J. Lu, J. Tu and L.Z. Shi, Linear differential equations with coefficients of (p,q)-order in the complex plane, J.Math.Anal.Appl, 372 (2010), 55-67. https://doi.org/10.1016/j.jmaa.2010.05.014
  16. X. Shen, J. Tu and H. Y. Xu, Complex oscillation of a second-order linear differential equation with entire coefficients of [p, q]-φ order, Advances in Difference Equations 2014 (2014), 2014:200.
  17. D. Shea, L.R. Sons. Value distribution theory for meromorphic functions of slow growth in the disc, Houston J. Math. 12 (2) (1986), 249-266.
  18. L.R. Sons, Unbounded functions in the unit disc, Internet. J. Math. Math. Sci., 6 (2) (1983), 201-242. https://doi.org/10.1155/S0161171283000204
  19. M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, Nwe York, 1975, reprint of the 1959 edition.
  20. J. Tu, Y. Zeng, H.Y.Xu, The Order and Type of Meromorphic Functions and Entire Functions of Finite Iterated Order, J. Computational Analysis and Applications, 21 (2016), no.5, 994-1003.
  21. L. Yang, Value Distribution Theory and Its New Research, Science Press, Beijing, 1982 (in Chinese).
  22. H.X. Yi and C.C. Yang, The Uniqueness Theory of Meromorphic Function, Science Press, Beijing, 1995(in Chinese).