关于Tumura-Clunie定理的注记

Rearks on Tumura-Clunie Theorem

设f为非常数亚纯函数 ,F =anfn+an-kfn-k+… +a0 ,(1≤k为自然数 ) ,其中a0 ,a1,… ,an-k,an 为亚纯函数 ,满足T(r,ai(z) ) =o(T(r,f) ) ,r→∞ ,r E mesE<∞ ,那么有 若k≤ 2则F ≡an(f +an- 1nan) n 或 2T(r,f)≤N(r,1F) +N(r,f) +N(r,1f +an- 1nan) +S(r,f) , 若k≥ 3则F ≡anfn 或者N(r,1F) ≥ (k- 2 )T(r,f) +S(r ,f) .

Let f and a_0,a_1,…a_ n-k,a_n be monconstant meromorphic functio ns i n the complex plane, satisfy T(r,a_i)=S(r,f), where S(r,f)=o(T(r,f)) a s r→∞, possibly outside a set of finite Lebesgue measure. Let F=a_nf-n+ a_ n-kf- n-k+…+a_0,we have ?if k≤2, then F≡a_n(f+a _ n-1na_n)-n or 2T(r,f)≤N(r,1F)+N (r,f)+N(r,1f+a_ n-1na_n) +S(r,f).?if k≥3,then F≡a_nf-n or N(r,1F)≥ (k-2)T(r,f)+S(r,f).