A note on the value distribution of $f^2(f')^n$ for $n\geq 2$
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초록
Let $f$ be a transcendental meromorphic function in the complex plane $\mathbb{C}$, and $a$ be a nonzero constant. We give a quantitative estimate of the characteristic function $T(r,f)$ in terms of $N(r,1/(f^2(f')^n-a))$, which states as following inequality, for positive integers $n\geq 2$, $$T(r,f)\leq \left(3+\frac{6}{n-1}\right)N\left(r,\frac{1}{af^2(f')^n-1}\right)+S(r,f).$$
Abstract
Let $f$ be a transcendental meromorphic function in the complex plane $\mathbb{C}$, and $a$ be a nonzero constant. We give a quantitative estimate of the characteristic function $T(r,f)$ in terms of $N(r,1/(f^2(f')^n-a))$, which states as following inequality, for positive integers $n\geq 2$, $$T(r,f)\leq \left(3+\frac{6}{n-1}\right)N\left(r,\frac{1}{af^2(f')^n-1}\right)+S(r,f).$$
참고문헌 (13)
- A. Alotaibi, "On the zeros of aff(k) − 1", Complex Var. Theory Appl, 49(13), pp.977~989 (2004)
- A. Alotaibi, "On the zeros of af(f(k))n − 1 for n ≥ 2", Comput. Methods Funct. Theory, 4(1), pp.227~235 (2004)
- A. A. Goldberg, "Value distribution of meromorphic functions, vol. 236" (2008), American Mathematical Society
- W. K. Hayman, "Meromorphic Functions" (1964), Oxford University Press
- X. J. Huang, "On the value distribution of f2f(k)", J. Aust. Math. Soc, 78(1), pp.17~26 (2005)
- Y. Jiang, "A note on the value distribution of fl(f(k))n"
- I. Lahiri, "Inequalities arising out of the value distribution of a differential monomial", J. Inequal. Pure Appl. Math, 4(2), pp.6~ (2003)
- P. Li, "On the value distribution of a certain type of differential polyno-mials", Monatsh. Math, 125(1), pp.15~24 (1998)
- C. K. Tse, "On the value distribution of fl(f(k))n", Kodai Math. J, 17(1), pp.163~169 (1994)
- J. P. Wang, "On the zeros of fn(z)f(k)(z)−c(z)", Complex Var. Theory Appl, 48(8), pp.695~703 (2003)
- C. C. Yang, "On the value distribution of ff(k)", Kodai Math. J, 19(2), pp.157~167 (1996)
- C. C. Yang, "A unicity theorem for meromorphic functions with deficient values", Acta Math. Sinica, 37(1), pp.62~72 (1994)
- Q. D. Zhang, "A growth theorem for meromorphic function", J. Chengdu Inst. Meteor, 20, pp.12~20 (1992)