Previous |  Up |  Next

Article

Full entry | PDF   (0.2 MB)
Keywords:
meromorphic function; differential polynomial; small function; sharing
Summary:
The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not \equiv 0, \infty$) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and $(k+1)\overline N(r, \infty ; f)+ \overline N(r, 0; f')+ N_{k}(r, 0; f')< \lambda T(r, f')+ S(r, f')$ for some real constant $\lambda \in (0, 1)$, then $f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \not \equiv 0$.
References:
[1] Al-Khaladi, A. H. H.: On meromorphic functions that share one small function with their $k$th derivative. Result. Math. 57 (2010), 313-318. DOI 10.1007/s00025-010-0029-1 | MR 2651117 | Zbl 1198.30032
[2] Al-Khaladi, A. H. H.: On entire functions which share one small function CM with their $k$th derivative. Result. Math. 47 (2005), 1-5. DOI 10.1007/BF03323007 | MR 2129572 | Zbl 1074.30027
[3] Al-Khaladi, A. H. H.: On entire functions which share one small function CM with their first derivative. Kodai Math. J. 27 (2004), 201-205. DOI 10.2996/kmj/1104247345 | MR 2100917 | Zbl 1070.30012
[4] Brück, R.: On entire functions which share one value CM with their first derivative. Result. Math. 30 (1996), 21-24. DOI 10.1007/BF03322176 | MR 1402421
[5] Gundersen, G. G.: Meromorphic functions that share finite values with their derivative. J. Math. Anal. Appl. 75 (1980), 441-446. DOI 10.1016/0022-247X(80)90092-X | MR 0581831 | Zbl 0447.30018
[6] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs 14 Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[7] Lahiri, I., Sarkar, A.: Uniqueness of a meromorphic function and its derivative. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 5 (2004), Article No. 20, 9 pages. MR 2048496 | Zbl 1056.30030
[8] Liu, L., Gu, Y.: Uniqueness of meromorphic functions that share one small function with their derivatives. Kodai Math. J. 27 (2004), 272-279. DOI 10.2996/kmj/1104247351 | MR 2100923 | Zbl 1115.30034
[9] Mues, E., Steinmetz, N.: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen. Manuscr. Math. 29 German (1979), 195-206. DOI 10.1007/BF01303627 | MR 0545041 | Zbl 0416.30028
[10] Rubel, L. A., Yang, C. C.: Values shared by an entire function and its derivative. Complex Anal., Proc. Conf., Univ. Lexington, 1976 Lect. Notes Math. 599 Springer, Berlin (1977), 101-103 J. D. Buckholtz et al. DOI 10.1007/BFb0096830 | MR 0460640 | Zbl 0362.30026
[11] Yang, L.-Z.: Solution of a differential equation and its applications. Kodai Math. J. 22 (1999), 458-464. DOI 10.2996/kmj/1138044097 | MR 1727305 | Zbl 1004.30021
[12] Yang, L.-Z.: Entire functions that share finite values with their derivatives. Bull. Aust. Math. Soc. 41 (1990), 337-342. DOI 10.1017/S0004972700018190 | MR 1071033 | Zbl 0691.30022
[13] Yu, K.-W.: On entire and meromorphic functions that share small functions with their derivatives. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 4 (2003), Article No. 21, 7 pages. MR 1966001 | Zbl 1021.30030
[14] Zhang, Q. C.: The uniqueness of meromorphic functions with their derivatives. Kodai Math. J. 21 (1998), 179-184. DOI 10.2996/kmj/1138043871 | MR 1645603 | Zbl 0932.30027

Partner of