关于亚纯函数理论中与极点无涉的基本不等式

<正> 关于奈望利纳(Nevanlinna)氏第二基本不等式曾有引入纪(导)数而作之种种不同的推广,在这些推广式中,极点的密指量每见出现且有特殊作用,因之能否将此量消去是一问题.米约(Milloux)氏及熊庆来教授由不同途径各获得与极点无涉之一不等式此二结果形状互异而各有特点.熊庆来教授指出这两个结果尚可推广到更普遍的境地,我们由此方向探研得如下两个定理,为熊、米二氏者之推广.

ON THE FUNDAMENTAL INEQUALITIES WITHOUT THE INTERVENTION OF THE POLES

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,(?)= oT(r,f~(k))],T(r,ψ_2)=oT(r,f~(k)]andT(r,(?))=oT(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,(?))=oT(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