Normal families related to shared values
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摘要: 设\,$\F$\,为区域\,$D$\,上的亚纯函数族, $a$\,和\,$b$\,都是不为零的两个有穷复数($\frac{a}{b}$\,不是正整数), 若每个\,$f \in \F$, $f(z) = a \Rightarrow f^{(k)}(z) = a$, 且\,$f - a$\,的零点重级至少为\,$k$, 当\,$f^{(k)}(z) = b$\,时有\,$|f(z) - a| \geq \varepsilon$, 其中\,$\varepsilon$\,为正数. 则\,$\F$\,在区域\,$D$\,内正规.Abstract: Let $\F$ be a family of meromorphic functions on a domain $D$, $a$ and $b$ be two nonzero finite complex numbers($\frac{a}{b}$ not positive integer). If for every $f \in \F$, $f(z) = a \Rightarrow f^{(k)}(z) = a$, and the zeros multiplicity of $f - a$ is at least $k$, and $|f(z) - a| \geq \varepsilon$ ($\varepsilon 0)$ whenever $f^{(k)}(z) = b$, then $\F$ is normal on $D$.
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Key words:
- meromorphic function /
- zero point /
- normality
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[1] {1}SCHWICK W. Sharing values and normality[J]. Arch Math, 1992, 59:50-54.{2}PANG X C, ZALCMAN L. Normality and shared values[J]. Ark Mat, 2000,38: 171-182.{3}LIN W C, YI H X. Value distribution of meromorphic functionconcerning shared values[J]. Indian J Pure Appl Math, 2003, 34:535-541.{4}FANG M L, ZALCMAN L. A note on normality and shared values[J]. JAust Math Soc, 2004, 76: 141-150.{5}PANG X C, ZALCMAN L. Normal families and shared values[J]. BullLondon Math Soc, 2000, 32: 325-331.{6}HAYMAN W K. Meromorphic Functions[M]. Oxford: Clarendon Press, 1964. -
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