【摘要】 在本文中,亚纯函数是指在整个复平面上的亚纯函数.本文是利用复分析的值分布理论来研究亚纯函数的唯一性.设f(z)和g(z)是两个亚纯函数,当fn(z)f′(z),gn(z)g′(z)分担1或者z CM时,前人给出了下面的定理:定理A设f(z)和g(z)是两个非常数亚纯函数,n≥11是一个正整数,如果fn(z)f′(z),gn(z)g′(z)分担1CM,则f(z)=c1ecz,g(z)=c2e-cz,这里c1,c2和c是3个常数且满足(c1c2)n+1c2≡-1;或者f(z)≡tan(z)其中t是一个常数满足tn+1=1.定理B设f(z)和g(z)是两个非常数亚纯函数(整函数),n≥11(n≥6)是一个正整数,如果fn(z)f′(z),gn(z)g′(z)分担z CM,则f(z)=c1ecz2,g(z)=c2e-cz2,这里c1,c2和c是3个常数且满足4(c1c2)n+1c2≡-1;或者f(z)≡tan(z)其中t是一个常数满足tn+1=1.在本文中,我们推广了上述定理,证明了下面的结论:设p(z)为n1次多项式,f(z)和g(z)是两个超越亚纯函数,n≥max{11,2n1+2}是一个正整数,如果fn(z)f′(z),gn(z)g′(z)分担多项式p(z)CM,则f(z)=c1e∫cp(z)dz,g(z)=c2e-∫cp(z)dz,这里c1,c2和c是3个常数且满足(c1c2)n+1c2≡-1;或者f(z)≡tan(z)其中t是一个常数满足tn+1=1.更多还原

【Abstract】 In this paper,by a meromorphic function we always mean a function which is meromorphic in the whole complex plane C.We use the theory of value distribution and study the uniqueness of meromorphic functions.Let f(z) be meromorphic function.We use the standard notations in Nevanlinna’s value distribution theory of meromorphic functions such as T(r,f),N(r,f),and m(r,f).The notation S(r,f) is defined to any quantity satisfying S(r,f)=o(T(r,f)) as r→∞,possibly outside of a set of rof finite linear measure.A meromorphic function a(z) is called a small function with respect to f(z) provided that T(r,a)=S(r,f).Let f(z) and g(z) be two meromorphic functions,and let a(z) be a small function with respect to f(z) and g(z).We say that two meromorphic functions f(z) and g(z) share a(z) IM(ignoring multiplicities) when f(z)-a(z) and g(z)-a(z) have the same zeros.If f(z)-a(z) and g(z)-a(z) have the same zeros with the same multiplicities,then we say that f(z) and g(z) share a(z) CM(counting multiplicities).Let f(z) and g(z) be two meromorphic functions.If fn(z)f′(z) and gn(z)g′(z) share 1 or z CM,previous workers have obtained the following theorems:Theorem A Let f(z) and g(z) be two nonconstant meromorphic functions and n≥11 a positive integer.If fn(z)f′(z) and gn(z)g′(z) share 1 CM,then either f(z)=c1ecz,g(z)=c2e-cz where c1,c2 and c are three constants satisfying(c1c2)n+1c2=-1,or f(z)≡tan(z) for a constant t such that tn+1=1.Theorem B Let f(z) and g(z) be two nonconstant meromorphic(entire) functions and n≥11(n≥6) a positive integer.If fn(z)f′(z) and gn(z)g′(z) share z CM,then either f(z)=c1ecz2,g(z)=c2e-cz2,where c1,c2 and c are three constants satisfying 4(c1c2)n+1c2=-1,or f(z)≡tan(z) for a constant t such that tn+1=1.In this paper,we will extend the above results as follows.Let f(z) and g(z) be two transcendental meromorphic functions,p(z) a polynomial of degree n1,and n≥max{11,2n1+2} a positive integer.If fn(z)f′(z) and gn(z)g′(z) share p(z) CM,then either f(z)=c1ec∫p(z)dz,g(z)=c2e-c∫p(z)dz,where c1,c2 and c are three constants satisfying(c1c2)n+1c2=-1,or f(z)≡tan(z) for a constant t such that tn+1=1.更多还原