Some generalizations of the unicity theory of Nevanlinna
In the present paper a generalization of R. Nevanlinna's five-point theorem is given: If \(f_ 1\) and \(f_ 2\) are transcendental meromorphic functions in the plane such that \(E(f_ 1=a_ j)=E(f_ 2=a_ j)\) for five distinct values \(a_ j\), then \(f_ 1=f_ 2\). Here, \(E(f=a)\) is simply the set of zeros of \(f-a\). This makes also sense if \(a(\not\equiv f)\) is also a meromorphic function. The generalization is as follows: If \(E(f_ 1=a_ j)=E(f_ 2=a_ j)\) for at least seven meromorphic functions \(a_ j\) satisfying \(T(r,a_ j)=O(\min (T(r,f_ 1),T(r,k)))\) then the same conclusion \(f_ 1=f_ 2\) holds. The same is true if seven is replaced by five, under the assumption that there exist two functions \(b_ 1\), \(b_ 2\) in the same class: \(T(r,b_ j)=O(\min (T(r,f_ 1),T(r,f_ 2))\) with large deficiency, \(\delta(f_ k,v_ k)>4/5\), \(k=1,2\).
scientific article; zbMATH DE number 488099
scientific article