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MSC2000 亚纯函数:6892,基本不等式:6580,甲3:1757,数字常数:1480,Nevanlinna:1336,公共零点:728,全平面:709,密指量:
ON THE FUNDAMENTAL INEQUALITIES WITHOUT THE INTERVENTION OF THE POLES
SHIEH HUI-CHUN(Fukien Normal College)
Abstract: H.Milloux and Professor K.L.Hiong had given two different inequalities whichdo not contain N(r,f)and which extend the fundamental inequality of Nevanlinna bydifferent methods.Professor K.L.Hiong has pointed out that we may extend the aboveresults to a general case.We obtain the following two theorems:Theorem I.Let f(x) and ψ_v(x)(v=1,2,3)be meromorphic functions,such thatfor r→∞T(r,(?)= o[T(r,f~(k))],T(r,ψ_2)=o[T(r,f~(k)]andT(r,(?))=o[T(r,f~(k)],and the ψ_v are different from one another.If (?)(O)≠∞;f(O)≠O,∞,(?)(O)andf~(K)(O)≠ψ_1~((k))(O),ψ_2(O),then the inequality(?)is satisfied for|x|=γ< p except,in case when ψ_v is of infinite order,a sequence ofthe intervals that their total lenagth is finite.The remainder S_k satisfies the conditions ofNevanlinna.Theorem Ⅱ.Let f(x) and ψ_v (x)(v=1,2,3)be meromorphic functions,such.that for γ→∞T(r,(?))=o[T(γ,f)],(v=1,2,3)and (?),(?)are distinct from(?)and do not reduce to zero.If(?)(O)≠∞,(?)(O)——(?)(O)=0,f(O)≠O,∞;F(O)≠0,1 and F′(O)≠0(?)then the inequality(?)is satisfied for |x|=γ

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DOI: cnki:ISSN:0583-1431.0.1959-03-004

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