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RADIAL OSCILLATION OF LINEAR DIFFERENTIAL EQUATION

  • Wu, Zhaojun (School of mathematics and statistics Hubei University of Science and Technology)
  • Received : 2011.04.29
  • Published : 2012.09.30

Abstract

In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

References

  1. S. B. Bank and I. Laine, On the oscillation theory of f'' + Af = 0 where A is entire, Trans. Amer. Math. Soc. 273 (1982), no. 1, 351-363.
  2. S. B. Bank and J. K. Langley, Oscillation theory for higher order linear differential equations with entire coefficients, Complex Variables Theory Appl. 16 (1991), no. 2-3, 163-175. https://doi.org/10.1080/17476939108814478
  3. L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differ- ential equation, Proc. Amer. Math. Soc. 101 (1987), no. 2, 317-322.
  4. T.-B. Cao, Complex oscillation of entire solutions of higher-order linear differential equations, Electron. J. Differential Equations 2006 (2006), no. 81, 8 pp. (electronic).
  5. C. Chuang, Differential Polynomials of Meromorphic Functions, Beijing Normal University Press, Beijing, 1999.
  6. S. Gao, Z. Chen, and T. Chen, Oscillation Theory of Linear Differential Equation, Huazhong University of Science and Technology Press, Wuhan, 1998.
  7. Z. Huang and Z. Chen, Angular distribution with hyper-order in complex oscillation theory, Acta Math. Sinica (Chin. Ser.) 50 (2007), no. 3, 601-614.
  8. Z. Huang and Z. Chen, The zeros distribution of solutions of higher order differential equations in an angular domain, Bull. Korean Math. Soc. 47 (2010), no. 3, 443-454. https://doi.org/10.4134/BKMS.2010.47.3.443
  9. Z. Huang and Z. Chen, The zeros distribution of hyper solutions of higher order differential equations in angular domain, Southeast Asian Bull. Math. 34 (2010), no. 3, 429-438.
  10. G. Jank and L. Volkmann, Meromorphe funktionen und Differentialiagleichungen, Birkhauser, 1985.
  11. L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. 22 (1998), no. 4, 385-405.
  12. I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15. Walter de Gruyter Co., Berlin, 1993.
  13. R. Nevanlinna, uber eine Klasse meromorpher Funktionen, Math. Ann. 92 (1924), no. 3-4, 145-154. https://doi.org/10.1007/BF01448004
  14. J. Rossi and S. Wang, The radial oscillation of solutions to ODE's in the complex domain, Proc. Edinburgh Math. Soc. (2) 39 (1996), no. 3, 473-483. https://doi.org/10.1017/S0013091500023233
  15. D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc. 69 (1963), 411-414. https://doi.org/10.1090/S0002-9904-1963-10951-9
  16. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Co. Ltd, Tokyo, 1959.
  17. S. Wang, On the sectorial oscillation theory of f'' + A(z)f = 0, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes (1994), no. 92, 66 pp.
  18. S. Wu, Angular distribution in complex oscillation, Sci. China Ser. A 48 (2005), no. 1, 107-114. https://doi.org/10.1360/03YS0159
  19. S. Wu, On the location of zeros of solution of f'' +Af = 0 where A(z) is entire, Math. Scand. 74 (1994), no. 2, 293-312.
  20. Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338. https://doi.org/10.4134/JKMS.2007.44.6.1329
  21. Z. Wu and D. Sun, Sectorial oscillation of linear differential equations and iterated order, Electron. J. Differential Equations 2007 (2007), no. 134, 7 pp. (electronic).
  22. L. Yang, Value Distribution Theory, Translated and revised from the 1982 Chinese original. Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1993.
  23. C. Yi, The augular distribution of the solutions of higher order differential equation, Acta Math. Sinica (Chin. Ser.) 48 (2005), no. 1, 133-140.
  24. J. Zheng, Value Distrbution of Meromorphic Functions, Springer, Beijing: Tsinghua Univ. Press. 2010.

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  1. Sectorial oscillation theory of linear differential equations vol.33, pp.4, 2013, https://doi.org/10.1016/S0252-9602(13)60072-7