设m(≥0)是一个正整数,h(z)(≠0)是区域D内的全纯函数,且其零点重级均≤m,P是多项式满足deg P≥3,或者degP=2且P仅有一个零点.设F是区域D内的一族亚纯函数,其零点与极点重级均≥m+1.如果对于F中的任意两个函数f,g,P(f)f′与P(g)g′分担h...设m(≥0)是一个正整数,h(z)(≠0)是区域D内的全纯函数,且其零点重级均≤m,P是多项式满足deg P≥3,或者degP=2且P仅有一个零点.设F是区域D内的一族亚纯函数,其零点与极点重级均≥m+1.如果对于F中的任意两个函数f,g,P(f)f′与P(g)g′分担h(z),则F在区域D内正规.该结果改进了Lei and Fang^([8]),Zhang^([16])等人的结果.展开更多
研究了亚纯函数与其差分算子分担多项式的唯一性问题,证明了:设f是一个有穷级非常数亚纯函数,p(z)(■0)是一个多项式.如果f,△_cf与△_c^2f CM分担∞,p(z),则f≡△_cf或f(z)=e^(Az+B)+b,其中p(z)≡b≠0,A≠0满足e^(Ac)=1.本文结果是对Ch...研究了亚纯函数与其差分算子分担多项式的唯一性问题,证明了:设f是一个有穷级非常数亚纯函数,p(z)(■0)是一个多项式.如果f,△_cf与△_c^2f CM分担∞,p(z),则f≡△_cf或f(z)=e^(Az+B)+b,其中p(z)≡b≠0,A≠0满足e^(Ac)=1.本文结果是对Chang, Fang(Chang J M, Fang M L. Uniqueness of entire functions and fixed points [J]. Kodai Math J, 2002, 25(1):309-320.)结果的差分模拟,并且完整回答了Chen, Chen(Chen B Q, Chen Z X, Li S. Uniqueness theorems on entire functions and their difference operators or shifts [J]. Abstr Appl Anal, 2012,Art. ID 906893, 8 pp.)的问题.展开更多
In this article,two results concerning the periodic points and normality of meromorphic functions are obtained:(i)the exact lower bound for the numbers of periodic points of rational functions with multiple fixed poin...In this article,two results concerning the periodic points and normality of meromorphic functions are obtained:(i)the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by let ting R(z)be a nonpolynomial rational function,and if all zeros and poles of R(z)—z are multiple,then Rk(z)has at least k+1 fixed points in the complex plane for each integer k≥2;(ii)a complete solution to the problem of normality of meromorphic functions with periodic points is given by let ting F be a family of meromorphic functions in a domain D,and let ting k≥2 be a positive integer.If,for each f∈F,all zeros and poles of f(z)-z are multiple,and its iteration fk has at most k distinct fixed points in D,then F is normal in D.Examples show that all of the conditions are the best possible.展开更多
In this paper,we mainly discuss entire solutions of finite order of the following Fermat type differential-difference equation[f(k)(z)]2+[△cf(z)]2=1,and the systems of differential-difference equations of the from ■...In this paper,we mainly discuss entire solutions of finite order of the following Fermat type differential-difference equation[f(k)(z)]2+[△cf(z)]2=1,and the systems of differential-difference equations of the from ■Our results can be proved to be the sufficient and necessary solutions to both equation and systems of equations.展开更多
文摘设m(≥0)是一个正整数,h(z)(≠0)是区域D内的全纯函数,且其零点重级均≤m,P是多项式满足deg P≥3,或者degP=2且P仅有一个零点.设F是区域D内的一族亚纯函数,其零点与极点重级均≥m+1.如果对于F中的任意两个函数f,g,P(f)f′与P(g)g′分担h(z),则F在区域D内正规.该结果改进了Lei and Fang^([8]),Zhang^([16])等人的结果.
文摘研究了亚纯函数与其差分算子分担多项式的唯一性问题,证明了:设f是一个有穷级非常数亚纯函数,p(z)(■0)是一个多项式.如果f,△_cf与△_c^2f CM分担∞,p(z),则f≡△_cf或f(z)=e^(Az+B)+b,其中p(z)≡b≠0,A≠0满足e^(Ac)=1.本文结果是对Chang, Fang(Chang J M, Fang M L. Uniqueness of entire functions and fixed points [J]. Kodai Math J, 2002, 25(1):309-320.)结果的差分模拟,并且完整回答了Chen, Chen(Chen B Q, Chen Z X, Li S. Uniqueness theorems on entire functions and their difference operators or shifts [J]. Abstr Appl Anal, 2012,Art. ID 906893, 8 pp.)的问题.
基金supported by the NNSF of China(11901119,11701188)The third author was supported by the NNSF of China(11688101).
文摘In this article,two results concerning the periodic points and normality of meromorphic functions are obtained:(i)the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by let ting R(z)be a nonpolynomial rational function,and if all zeros and poles of R(z)—z are multiple,then Rk(z)has at least k+1 fixed points in the complex plane for each integer k≥2;(ii)a complete solution to the problem of normality of meromorphic functions with periodic points is given by let ting F be a family of meromorphic functions in a domain D,and let ting k≥2 be a positive integer.If,for each f∈F,all zeros and poles of f(z)-z are multiple,and its iteration fk has at most k distinct fixed points in D,then F is normal in D.Examples show that all of the conditions are the best possible.
基金supported by the National Natural Science Foundation of China(11701188)
文摘In this paper,we mainly discuss entire solutions of finite order of the following Fermat type differential-difference equation[f(k)(z)]2+[△cf(z)]2=1,and the systems of differential-difference equations of the from ■Our results can be proved to be the sufficient and necessary solutions to both equation and systems of equations.
