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Meromorphic functions sharing three values or sets CM. (English) Zbl 0930.30027

In addition to the usual terminology of Nevanlinna theory, let \(\overline{N}_{k)}(r,1/(f-a))\) and \(\overline{N}_{(k}(r,1/(f-a))\) be the reduced counting functions of the \(a\)-points of a meromorphic function \(f\) which have multiplicity at most \(k\) or at least \(k\), respectively. It is shown in Theorem 1 that if \(f\) and \(g\) are meromorphic functions which share 0,1,\(\infty\) counting multiplicity and which are not Möbius transformations of each other, and if there exists \(a\not=0,1,\infty\) such that \({T(r,f)\leq c\overline{N}_{(2}(r,{{1}\over{f-a}})+S(r,f)}\) for some \(c>0\), then \(f\) and \(g\) have the form \({f={{{e^{t\gamma}-1}\over{\lambda e^{-s\gamma}-1}}}, g={{{e^{-t\gamma}-1}\over{{{1}\over{\lambda}} e^{s\gamma}-1}}}}\) where \(\gamma\) is a non-constant entire function, \(\lambda\) a non-zero constant and \(s\) and \(t\) are mutually prime integers, \(t>0\). Moreover, \(\theta:=-t/s\not=1,a\) and \({{{(1-a)^{s+t}}\over{a^t}}=\lambda^t{{(1-\theta)^{s+t}}\over{\theta^t}}}\). Theorem 2 gives a result of this type if \({\overline{N}_{1)}(r,{{1}\over{f-a}})=S(r,f)}\). Here \(f\) and \(g\) have one of nine possible forms, which are listed explicitly. Theorem 3 gives a result for functions sharing sets.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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