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Meromorphic functions sharing one value with their derivatives concerning the difference operator. (English) Zbl 1383.30010

Summary: Let \(f(z)\) be a non-constant meromorphic function of finite order, \(c\in \mathbb {C}\setminus \{0\}\) and \(k\in \mathbb {N}\). Suppose \(f(z)\) and \(f^{(k)}(z+c)\) share 1 CM (IM), \(f(z)\) and \(f(z+c)\) share \(\infty \) CM. If \(N(r,0;f)=S(r,f) \left( N\left( r,0;f(z)\right) +N\left( r,0;f^{(k)}(z+c)\right) =S(r,f)\right) \), then either \(f(z)\equiv f^{(k)}(z+c)\) or \(f(z)\) is a solution of the following equation: \[ \begin{aligned} &f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right),\text{ and}\\&N\left( r,0;f(z)+\frac{1}{a(z)}\right) =S(r,f)\\ &\left( f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right) \right) \end{aligned} \] where \(a(z)\left( \not \equiv -\,1,0,\infty \right) \left( a(z)\left( \not \equiv 0,\infty \right) \right) \) is a meromorphic function satisfying \(T(r,a)=S(r,f)\). Also we exhibit some examples to show that the conditions of our results are the best possible.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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[1] Al-Khaladi, A.H.H.: Meromorphic functions that share one finite value CM or IM with their first derivative. J AL-Anbar Univ. Pure Sci. 3, 69-73 (2009)
[2] Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic \[f(z+\eta )\] f(z+η) and difference equations in complex plane. Ramanujan J. 16, 105-129 (2008) · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1
[3] Frank, G., Weissenborn, G.: Meromorphe Funktionen, die mit einer ihrer Ableitungen Werte teilen. Complex Var. 7, 33-43 (1986) · Zbl 0603.30038
[4] Gundersen, G.G.: Meromorphic functions that share finite values with their derivative. J. Math. Anal. Appl. 75, 441-446 (1980) · Zbl 0447.30018 · doi:10.1016/0022-247X(80)90092-X
[5] Gundersen, G.G.: Meromorphic functions that share two finite values with their derivative. Pacific J. Math. 105, 299-309 (1983) · Zbl 0497.30025 · doi:10.2140/pjm.1983.105.299
[6] Hayman, W.K.: Meromorphic Functions. The Clarendon Press, Oxford (1964) · Zbl 0115.06203
[7] Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J., Zhang, J.L.: Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 352-363 (2009) · Zbl 1180.30039 · doi:10.1016/j.jmaa.2009.01.053
[8] Li, P., Yang, C.C.: Value sharing of an entire function and its derivatives. J. Math. Soc. Jpn. 51, 781-799 (1999) · Zbl 0938.30023 · doi:10.2969/jmsj/05140781
[9] Mues, E., Steinmetz, N.: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen. Manuscr. Math. 29, 195-206 (1979) · Zbl 0416.30028 · doi:10.1007/BF01303627
[10] Rubel, L.A., Yang, C.C.: Values shared by an entire function and its derivative, Lecture Notes in Math., Vol. 599, Springer, New York, pp. 101-103 (1977) · Zbl 0362.30026
[11] Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht (2003) · Zbl 1070.30011 · doi:10.1007/978-94-017-3626-8
[12] Zhang, J.L., Yang, L.Z.: Some results related to a conjecture of R. Brück, J. Inequal. Pure Appl. Math., 8, Art. 18 (2007) · Zbl 1136.30009
[13] Zheng, J.H., Wang, S.P.: On unicity properties of meromorphic functions and their derivatives. Adv. Math. 21, 334-341 (1992) · Zbl 0783.30026
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