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Normality and shared values. (English) Zbl 1079.30044
For $f$ meromorphic on the unit disc $\Delta$ and $a\in \Bbb{C}$ define $\overline E_f(a)=f^{-1}(\{a\})\cap\Delta=\{\,z\in\Delta:f(z)=a\,\}$. Two functions $f$ and $g$ on $\Delta$ are said to share the value $a$ if $\overline E_f(a)=\overline E_g(a)$. A meromorphic function $f$ on $ \Bbb{C}$ is called a normal function if there exists a positive number $M$ such that $f^\#(z)\le M$, where $f^\#(z)=\vert f'(z)\vert /(1+\vert f(z)\vert ^2)$ denotes the spherical derivative. The authors prove the following theorems: 1) Let $\Cal{F}$ be a family of meromorphic functions on the unit disc $\Delta$, and let $a$ and $b$ be distinct complex numbers and $c$ a nonzero complex number. If for every $f\in \Cal{F}$, $\overline E_f(0)=\overline E_{f'}(a)$, $\overline E_f(c)=\overline E_{f'}(b)$, then $\Cal{F}$ is normal on $\Delta$. Earlier a similar result has been proved by {\it W. Schwick} [Arch. Math. 59, No. 1, 50--54 (1992; Zbl 0758.30028)]. 2) Let $f$ be a meromorphic function on $\Bbb{C}$ and $a$ and $b$ be distinct complex numbers. If $f$ and $f'$ share $a$ and $b$, then $f$ is a normal function. This should be compared with the result of {\it E. Mues} and {\it N. Steinmetz} in [Manuscr. Math. 29, 195--206 (1979; Zbl 0416.30028)].

30D45Bloch functions, normal functions, normal families
Full Text: DOI
[1] Bergweiler, W. andEremenko, A., On the singularities of the inverse to a meromorphic function of finite order,Rev. Mat. Iberoamericana 11 (1995), 355--373. · Zbl 0830.30016
[2] Chen, H. andGu, Y., An improvement of Marty’s criterion and its applications,Sci. China Ser. A 36 (1993), 674--681.
[3] Clunie, J. andHayman, W. K., The spherical derivative of integral and meromorphic functions,Comment. Math. Helv. 40, (1966), 117--148. · Zbl 0142.04303 · doi:10.1007/BF02564366
[4] Frank, G. andWeissenborn, G., Rational deficient functions of meromorphic functions,Bull. London Math. Soc. 18 (1986), 29--33. · Zbl 0586.30025 · doi:10.1112/blms/18.1.29
[5] Hayman, W. K., Picard values of meromorphic functions and their derivatives,Ann. of Math. 70 (1959), 9--42. · Zbl 0088.28505 · doi:10.2307/1969890
[6] Hayman, W. K.,Meromorphic Functions, Clarendon Press, Oxford, 1964.
[7] Minda, D., Yosida functions, inLectures on Complex Analysis (Chuang, C.-T., ed.), pp. 197--213, World Scientific Publ., Singapore, 1988. · Zbl 0744.30022
[8] Mues, E. andSteinmetz, N., Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen,Manuscripta Math. 29 (1979), 195--206. · Zbl 0416.30028 · doi:10.1007/BF01303627
[9] Pang, X., On normal criterion of meromorphic functions,Sci. China Ser. A 33 (1990), 521--527. · Zbl 0706.30024
[10] Pang, X., Shared values and normal families, Preprint, 1998. · Zbl 1030.30030
[11] Pang, X. andZalcman, L., Normal families and shared values, to appear inBull. London Math. Soc.
[12] Schwick, W., Sharing values and normality,Arch. Math. (Basel) 59 (1992), 50--54. · Zbl 0758.30028
[13] Yang, L.,Value Distribution Theory, Springer-Verlag, Berlin-Heidelberg, 1993. · Zbl 0790.30018