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Normal families of meromorphic functions and shared values. (English) Zbl 1253.30047

In [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)], W. Schwick first proved a normality criterion for a family of meromorphic functions which involves value sharing. This result inspired a number of researchers to establish normality criteria using the concept of value sharing. In the paper under review the authors prove two normality criteria for a family of meromorphic functions when \(P(f)f^{(k)}\) and \(P(g)g^{(k)}\) share a nonzero value for every pair of functions \(f\), \(g\) of the family, where \(P\) is a polynomial having some special properties.

MSC:

30D45 Normal functions of one complex variable, normal families
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0758.30028
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References:

[1] Clunie J., Hayman W.K.: The spherical derivative of integral and meromorphic functions. Comment Math. Helv. 40, 117–148 (1966) · Zbl 0142.04303
[2] Fang M.L., Zalcman L.: A note on normality and shared values. J. Aust. Math. Soc. 76, 141–150 (2004) · Zbl 1074.30032
[3] Grael J., Nevo S.: On a result of Singh and Singh concerning shared values and normality. Complex Var. Elliptic Equ. 55(4), 347–356 (2010) · Zbl 1188.30045
[4] Hayman W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964) · Zbl 0115.06203
[5] Hu P.C., Meng D.W.: Normality criteria of meromorphic functions with multiple zeros. J. Math. Anal. Appl. 357, 323–329 (2009) · Zbl 1175.30030
[6] Huang X.J., Gu Y.X.: On the value distribution of f 2 f (k). J. Aust. Math. Soc. 78, 17–26 (2005) · Zbl 1069.30050
[7] Lei C.L., Fang M.L.: Normality and shared values concerning differential polynomial. Sci. China Math. 53, 749–754 (2010) · Zbl 1213.30055
[8] Pang X.C., Zalcman L.: On theorems of Hayman and Clunie. N.Z. J. Math. 28, 71–75 (1999) · Zbl 0974.30025
[9] Pang X.C., Zalcman L.: Normal families and shared values. Bull. Lond. Math. Soc. 32, 325–331 (2000) · Zbl 1030.30031
[10] Schiff J.: Normal Families. Springer, Berlin (1993) · Zbl 0770.30002
[11] Schwick W.: Sharing values and normality. Arch. Math. 59, 50–54 (1992) · Zbl 0758.30028
[12] Steinmetz N.: Über die nullstellen von differentialpolynomen. Math. Z. 176, 255–264 (1981) · Zbl 0466.30026
[13] Xu Y., Qiu H.L.: An improvement of Montel’s criterion. J. Math. Anal. Appl. 343, 1075–1079 (2008) · Zbl 1167.30013
[14] Xu Y.: On a result due to Yang and Schwick (in Chinese). Sci. Sin. Math. 40(5), 421–428 (2010)
[15] Yang C.C., Hu P.C.: On the value distribution of ff (k). Kodai. Math. 19, 157–167 (1996) · Zbl 0867.30023
[16] Yang L.: Value distribution theory. Springer, Berlin (1993) · Zbl 0790.30018
[17] Zhang Q.C.: Some normality criteria of meromorphic functions. Complex Var. Elliptic Equ. 53, 791–795 (2008) · Zbl 1152.30033
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