涉及导函数与分担亚纯函数的正规定则
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摘要
针对涉及导函数与分担亚纯函数的正规定则,得到了如下结果:设Ω是区域D内的亚纯函数族,a(z)(≠0)是亚纯函数。若对于任意μ(z)∈Ω满足如下条件:(1)μ(z)≠0;(2)对μ(z)和a(z)的任意公共极点,其在μ(z)中的重级大于或等于在a(z)中的重级;(3)对任意函数对{μ(z),ν(z)}?Ω,μ~((m))(z)和ν~((m))(z)分担a(z),则Ω在D内正规。同时,给出了2个例子来说明条件(1)和(2)的必要性。
Aiming at normal criteria concerning derivative functions and shared meromorphic functions,the result is show ed as follow s. Let Ω and a( z)( ≠0) be a family of functions meromorphic and a meromorphic function in D,respectively. If every μ( z) ∈Ω satisfies conditions as follows:( 1) μ( z) ≠0,( 2) for each of the same poles of μ( z) and a( z),the multiplicities in μ( z) are greater than or equal to the multiplicities in a( z),( 3) for every functions { μ( z),ν( z) } ?Ω,μ~(( m))( z) and ν~(( m))( z) share a( z),then Ω is normal in D. And tw o examples are given to verify the necessity of conditions( 1) and( 2).
Aiming at normal criteria concerning derivative functions and shared meromorphic functions,the result is show ed as follow s. Let Ω and a( z)( ≠0) be a family of functions meromorphic and a meromorphic function in D,respectively. If every μ( z) ∈Ω satisfies conditions as follows:( 1) μ( z) ≠0,( 2) for each of the same poles of μ( z) and a( z),the multiplicities in μ( z) are greater than or equal to the multiplicities in a( z),( 3) for every functions { μ( z),ν( z) } ?Ω,μ~(( m))( z) and ν~(( m))( z) share a( z),then Ω is normal in D. And tw o examples are given to verify the necessity of conditions( 1) and( 2).
引文
[1] SCHWICK W. Sharing values and normality[J]. Archiv der Mathematik, 1992, 59(1):50-54.
[2] PANG X C, ZALCMAN L. Normality and shared values[J]. Arkiv f?r Matematik, 2000, 38(1):171-182.
[3] HAYMAN W K. Meromorphic functions[M]. Oxford: Clarendon Press, 1964.
[4] YANG L. Value distribution theory, translated and revised from the 1982 Chinese original[M]. Berlin: Springer, 1993.
[5] YANG C C, YI H X. Uniqueness theory of meromorphic functions[M].[S.l.]: Springer Science & Business Media, 2004.
[6] PANG X C, NEVO S, ZALCMAN L. Derivatives of meromorphic functions with multiple zeros and rational functions[J]. Computational Methods and Function Theory, 2008, 8(2):483-491.
[7] XU Y. On a result due to Yang and Schwick[J]. Sci Sin Math, 2010, 40(5):421-428.
[8] SCHWICK W. On Hayman?s alternative for families of meromorphic functions[J]. Complex Variables and Elliptic Equations, 1997, 32(1):51-57.
[9] PANG X C, ZALCMAN L. Normal families and shared values[J]. Bulletin of the London Mathematical Society, 2000, 32(3):325-331.
[10] HAYMAN W K. Picard values of meromorphic functions and their derivatives[J]. Annals of Mathematics, 1959(1):9-42.
[11] CHEN Q Y, QI J M. Criteria of normality concerning the sequence of omitted functions[J]. Bulletin of the Korean Mathematical Society, 2016, 53(5):1373-1384.
[2] PANG X C, ZALCMAN L. Normality and shared values[J]. Arkiv f?r Matematik, 2000, 38(1):171-182.
[3] HAYMAN W K. Meromorphic functions[M]. Oxford: Clarendon Press, 1964.
[4] YANG L. Value distribution theory, translated and revised from the 1982 Chinese original[M]. Berlin: Springer, 1993.
[5] YANG C C, YI H X. Uniqueness theory of meromorphic functions[M].[S.l.]: Springer Science & Business Media, 2004.
[6] PANG X C, NEVO S, ZALCMAN L. Derivatives of meromorphic functions with multiple zeros and rational functions[J]. Computational Methods and Function Theory, 2008, 8(2):483-491.
[7] XU Y. On a result due to Yang and Schwick[J]. Sci Sin Math, 2010, 40(5):421-428.
[8] SCHWICK W. On Hayman?s alternative for families of meromorphic functions[J]. Complex Variables and Elliptic Equations, 1997, 32(1):51-57.
[9] PANG X C, ZALCMAN L. Normal families and shared values[J]. Bulletin of the London Mathematical Society, 2000, 32(3):325-331.
[10] HAYMAN W K. Picard values of meromorphic functions and their derivatives[J]. Annals of Mathematics, 1959(1):9-42.
[11] CHEN Q Y, QI J M. Criteria of normality concerning the sequence of omitted functions[J]. Bulletin of the Korean Mathematical Society, 2016, 53(5):1373-1384.