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Further results on meromorphic functions that share two values with their derivatives. (English) Zbl 1073.30024

The authors extend results they obtained in [J. Math. Soc. Japan 51, No. 4, 781–799 (1999; Zbl 0938.30023)]. Let \(f\) be a nonconstant meromorphic function for which \(\overline N(r, f)= S(r, f)\) and \(L(f)= C_{n+1}f^{(n)}+ C_nf^{(n-1)}+\cdots+ c_2f'+ c_1f+ c_0\), where \(c_j\) (\(j= 0,1,2,\dots,n+ 1\))are small meromorphic functions of \(f\) and \(c_{n+1}\neq 0\). For a complex number \(a\), define \(\tau(a)\) to be \(\liminf_{r\to\infty}\,(\overline{N_0}(r,1/(f- a))/\overline N(r, 1/(f- a))\) when \(\overline N(r,1/f- a)\not\equiv 0\) and one otherwise, where \(\overline N_0(r,1/(f -a))\) denotes the counting function of those a points of \(f\) and \(L(f)\) of the same multiplicities, but counted only once. If for two distinct complex numbers \(a_1\) and \(a_2\), \(\max\{\tau(a_1),\tau(a_2)\} >1/2\), then the authors show \(f\) and \(L(f)\) assume one of four explicit forms.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0938.30023
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References:

[1] DOI: 10.2969/jmsj/05140781 · Zbl 0938.30023 · doi:10.2969/jmsj/05140781
[2] Hayman, Meromorphic functions (1964)
[3] DOI: 10.1016/0022-247X(80)90092-X · Zbl 0447.30018 · doi:10.1016/0022-247X(80)90092-X
[4] Bernstein, Forum Mathematicum 8 pp 379– (1996)
[5] Mues, Complex Variables 12 pp 167– (1989)
[6] Rubel, Values shared by an entire function and its derivative pp 101– (1977) · Zbl 0362.30026
[7] DOI: 10.1007/BF02565342 · JFM 52.0323.03 · doi:10.1007/BF02565342
[8] DOI: 10.1007/BF01303627 · Zbl 0416.30028 · doi:10.1007/BF01303627
[9] Wang, Pure Appl. Math.(Chinese) 17 pp 142– (2001)
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