Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Value Distribution of ff(k)

  • Wang, Jian-Ping (Department of Mathematics, Shaoxing College of Arts and Sciences)
  • Received : 2004.03.19
  • Published : 2006.06.23
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Abstract

This paper proves the following results: Let $f$ be a transcendental entire function, and let $k({\geq})2$ be a positive integer. If $T(r,\;f){\neq}N_{1)}(r,1/f)+S(r,\;f)$, then $ff^{(k)}$ assumes every finite nonzero value infinitely often. Also the case when f is a transcendental meromorphic function has been considered and some results are obtained.

Keywords

References

  1. W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11(1995), 355-373.
  2. H. H. Chen and M. L. Fang, On the value distribution of $f^n$f', Science in China, 38A(1995), 789-798.
  3. W. Doeringer, Exceptional Values of Differetial Polynomials, Pacific J. of Math., 98(1982), 55-62. https://doi.org/10.2140/pjm.1982.98.55
  4. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  5. W. K. Hayman, Research Problems in Function Theory, Athlone Press, London, 1967.
  6. W. K. Hayman, Picard value of meromorphic functions and their derivatives, Ann. Math., 70(1959), 9-42. https://doi.org/10.2307/1969890
  7. E. Mues, Uber ein problem von Hayman, Math. Z., 164(1979), 239-259. https://doi.org/10.1007/BF01182271
  8. X. C. Pang and L. Zalcmam, On theorems of Hayman and Clunie, New Zealand J. Math., 28(1999), 71-75.
  9. L. R. Sons, Deficiencies of monomials, Math. Z., 111(1969), 53-68. https://doi.org/10.1007/BF01110917
  10. N. Steinmetz, Uber die Nullstellen von Differentialpolynomen, Math. Z., 176(1981), 255-264. https://doi.org/10.1007/BF01261872
  11. Jian-Ping Wang, On the conjecture for zeros of the entire function $f^(k)$ f-a, Chin. Advances in Math., 31(2002), 41-46.
  12. C. C. Yang and P. C. Hu, On the value distribution of f $f^{(k)}$, Kodai Math. J., 19(1996), 157-167. https://doi.org/10.2996/kmj/1138043595
  13. C. C. Yang, L. Yang and Y. F. Wang, On zeros of $(f^{(k)})^n$ f-a, Chin. Sci. Bull., 38(1993), 2215-2218.
  14. H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Pure and Applied Math. Monography, Science Press, Beijing, 1995.
  15. L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc., 35(1998), 215-230. https://doi.org/10.1090/S0273-0979-98-00755-1