Normal families and shared values of meromorphic functions

Volume 80 / 2003

Mingliang Fang, Lawrence Zalcman Annales Polonici Mathematici 80 (2003), 133-141 MSC: Primary 30D45. DOI: 10.4064/ap80-0-11


Let ${\cal F}$ be a family of meromorphic functions on a plane domain $D,$ all of whose zeros are of multiplicity at least $k\ge 2.$ Let $a$, $b$, $c$, and $ d$ be complex numbers such that $d\not =b,0$ and $c\not =a.$ If, for each $f\in {\cal F},$ $f(z)=a\Leftrightarrow f^{(k)}(z)=b$, and $f^{(k)}(z)=d\Rightarrow f(z)=c,$ then ${\cal F}$ is a normal family on $D.$ The same result holds for $k=1$ so long as $b\not =(m+1)d,$ $m=1,2,\dots .$


  • Mingliang FangDepartment of Mathematics
    Nanjing Normal University
    Nanjing 210097, P.R. China
  • Lawrence ZalcmanDepartment of Mathematics and Statistics
    Bar-Ilan University
    52900 Ramat-Gan, Israel

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