摘要
设f(z)是超越半纯函数,并满足(r,f)+■(r,1/f)=S(r,f),我们用线性无关和局部展开的方法证明了:设ψ(Z)=α_np_N[f]+a_(n-1)P_(n-1)[f]+…+α_0是任一个f的微分多项式,并不退化为常数项,则ρ_ψ=ρ_f,μ_ψ=μ_f,且对任何复数C∈C,满足α_(n-1)P_(n-1)[f]+…+α_0-c■0都有θ(c,ψ)≤1-1/n。进而若ψ是齐次微分多项式,则对任何c∈C,c≠0都有θ(c,ψ)=0而且■(r,ψ)+■(r,1/ψ)=S(r,ψ),这里θ(c,ψ)=[(r,1/(ψ-c))/T(r,ψ)],P_i[f]是i次齐次微分多项式。
In thin paper,we consider differential polynomials of f,ψ=f^nQ[f]+P[f],where Q[f],P[f]are differential polynomials of f with Q[f]·P[f]0,and getthe following results:1.Let f be transcendental meromorphic,ψ=f^nQ[f]+P[f].If n≥Γ_P+1 and[((r,1/f)+(Γ_p—γ_p+1)(r,f)]/T(r,f)<n—γ_p,then ρ_Ψ=ρ_f and μ_Ψ=μ_f,where ρ,μ are order and lower order respectively.2.Let f be transcendental meromorphic and satisfy (r,f)+(r,1/f)=S(r,f).Suppose that ψ=a_nP_n[f]+α_(n-1)P_(n-1)[f]+…+α_0 is a differential polynomial of f,which is not a constant term.Then ρ_Ψ=ρ_f and μ_Ψ=μ_f,and (C,ψ)≤1—1/n forany C∈C satisfy a_(n-1)P_(n-1)[f]+…+α_0—C0.
出处
《晓庄学院自然科学学报》
CAS
1989年第3期206-211,共6页
Journal of Natural Science of Hunan Normal University
关键词
微分多项式
亏值
半纯函数
meromorphic function
differential algebra
deficiency
order/differential polynomials
