Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 305710, 7 pages
http://dx.doi.org/10.1155/2014/305710
Research Article

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Extremal Coefficients

1School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China

Received 21 December 2013; Accepted 25 March 2014; Published 10 April 2014

Academic Editor: Mohammad T. Darvishi

Copyright © 2014 Jianren Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964. View at MathSciNet
  2. I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993. View at MathSciNet
  4. C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557, Kluwer Academic Publishers, New York, NY, USA, 2003. View at MathSciNet
  5. G. G. Gundersen, “Finite order solutions of second order linear differential equations,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 415–429, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Hellerstein, J. Miles, and J. Rossi, “On the growth of solutions of f+gf+hf=0,” Transactions of the American Mathematical Society, vol. 324, no. 2, pp. 693–706, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. B. Bank and I. Laine, “On the oscillation theory of f+Af=0 where A is entire,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 351–363, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z.-X. Chen and C.-C. Yang, “Some further results on the zeros and growths of entire solutions of second order linear differential equations,” Kodai Mathematical Journal, vol. 22, no. 2, pp. 273–285, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. G. Gundersen, “On the question of whether f+ezf+B(z)f=0 can admit a solution f0 of finite order,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 102, no. 1-2, pp. 9–17, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Hellerstein, J. Miles, and J. Rossi, “On the growth of solutions of certain linear differential equations,” Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, vol. 17, no. 2, pp. 343–365, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. K.-H. Kwon and J.-H. Kim, “Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations,” Kodai Mathematical Journal, vol. 24, no. 3, pp. 344–351, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Ozawa, “On a solution of w+ezw+(az+b)w=0,” Kodai Mathematical Journal, vol. 3, no. 2, pp. 295–309, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z.-X. Chen and C.-C. Yang, “On the zeros and hyper-order of meromorphic solutions of linear differential equations,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 24, no. 1, pp. 215–224, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. K.-H. Kwon, “On the growth of entire functions satisfying second order linear differential equations,” Bulletin of the Korean Mathematical Society, vol. 33, no. 3, pp. 487–496, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K.-H. Kwon, “Nonexistence of finite order solutions of certain second order linear differential equations,” Kodai Mathematical Journal, vol. 19, no. 3, pp. 378–387, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. Yang, “Deficient values and angular distribution of entire functions,” Transactions of the American Mathematical Society, vol. 308, no. 2, pp. 583–601, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Wu, “Some results on entire functions of finite lower order,” Acta Mathematica Sinica. New Series, vol. 10, no. 2, pp. 168–178, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. J. R. Long, P. C. Wu, and Z. Zhang, “On the growth of solutions of second order linear differential equations with extremal coefficients,” Acta Mathematica Sinica, vol. 29, no. 2, pp. 365–372, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Belaïdi, “Growth and oscillation theory of non-homogeneous complex differential equations with entire coefficients,” Communications in Mathematical Analysis, vol. 5, no. 2, pp. 13–25, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Belaïdi and S. Abbas, “On the hyper order of solutions of a class of higher order linear differential equations,” Analele stiintifice ale Universitatii Ovidius Constanta, vol. 16, no. 2, pp. 15–30, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Chen, “On the hyper order of solutions of higher order differential equations,” Chinese Annals of Mathematics B, vol. 24, no. 4, pp. 501–508, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Z. Chen, “On the growth of sulutions of a class of higher order differential equations,” Acta Mathematica Scientia, vol. 24, pp. 52–60, 2004. View at Google Scholar
  23. J. Tu and C.-F. Yi, “On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 487–497, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. G. G. Gundersen, “Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates,” Journal of the London Mathematical Society. Second Series, vol. 37, no. 1, pp. 88–104, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. P. D. Barry, “Some theorems related to the πρ theorem,” Proceedings of the London Mathematical Society, vol. 21, pp. 334–360, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. B. Bank, “A general theorem concerning the growth of solutions of first-order algebraic differential equations,” Compositio Mathematica, vol. 25, pp. 61–70, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. L. Yang and G. H. Zhang, “Distribution of Borel directions of entire functions,” Acta Mathematica Sinica, vol. 19, no. 3, pp. 157–168, 1976 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, Japan, 1959. View at MathSciNet
  29. P. D. Barry, “On a theorem of Besicovitch,” The Quarterly Journal of Mathematics, vol. 14, pp. 293–302, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet