### A NOTE ON THE VALUE DISTRIBUTION OF f2(f')n FOR n≥2

Jiang, Yan

• 투고 : 2014.05.21
• 발행 : 2016.03.31
• 8 1

#### 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 $N(r,1/(f^2(f^{\prime})^n-a))$, which states as following inequality, for positive integers $n{\geq}2$, $$T(r,f){\leq}$3+{\frac{6}{n-1}}$N$r,{\frac{1}{af^2(f^{\prime})^n-1}}$+S(r,f)$$.

#### 키워드

transcendental meromorphic function;deficiency

#### 참고문헌

1. A. Alotaibi, On the zeros of $aff(^k)-1$, Complex Var. Theory Appl. 49 (2004), no. 13, 977-989.
2. A. Alotaibi, On the zeros of $af(f^{(k)})^n-1$ for $n{\geq}2$, Comput. Methods Funct. Theory 4 (2004), no. 1, 227-235. https://doi.org/10.1007/BF03321066
3. A. A. Goldberg and V. I. Ostrovskii, Value distribution of meromorphic functions, vol. 236, American Mathematical Society, Providence, RI, 2008.
4. W. K. Hayman, Meromorphic Functions, Oxford University Press, 1964.
5. X. J. Huang and Y. X. Gu, On the value distribution of $f^2f(^k)$, J. Aust. Math. Soc. 78 (2005), no. 1, 17-26. https://doi.org/10.1017/S1446788700015536
6. Y. Jiang and B. Huang, A note on the value distribution of $f^l(f^{(k)})^n$, in submission.
7. I. Lahiri and S. Dewan, Inequalities arising out of the value distribution of a differential monomial, J. Inequal. Pure Appl. Math. 4 (2003), no. 2, Article 27, 6 pp. (electronic).
8. P. Li and C. C. Yang, On the value distribution of a certain type of differential polyno-mials, Monatsh. Math. 125 (1998), no. 1, 15-24. https://doi.org/10.1007/BF01489455
9. C. K. Tse and C. C. Yang, On the value distribution of $f^l(f^{(k)})^n$, Kodai Math. J. 17 (1994), no. 1, 163-169. https://doi.org/10.2996/kmj/1138039905
10. J. P. Wang, On the zeros of $f^n(z)f^{(k)}(z)-c(z)$, Complex Var. Theory Appl. 48 (2003), no. 8, 695-703. https://doi.org/10.1080/0278107031000152607
11. C. C. Yang and P. C. Hu, On the value distribution of $ff^{(k)}$, Kodai Math. J. 19 (1996), no. 2, 157-167. https://doi.org/10.2996/kmj/1138043595
12. C. C. Yang and H. X. Yi, A unicity theorem for meromorphic functions with deficient values, Acta Math. Sinica 37 (1994), no. 1, 62-72.
13. Q. D. Zhang, A growth theorem for meromorphic function, J. Chengdu Inst. Meteor. 20 (1992), 12-20.