齐次与非齐次复线性复合函数方程亚纯解的增长性
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摘要
运用亚纯函数的Nevanlinna值分布理论,研究了一类齐次与非齐次复线性复合函数方程亚纯函数解的增长性,并推广至更一般的含微分的复线性复合函数方程的情形.当这些方程允许有多项系数具有最大级或最大下级时,在一定条件下得到了这些方程非零亚纯解的级或下级的下界的估计.
The growth of meromorphic solutions of a kind of homogenous and non-homogeneous complex linear equations for composite functions with meromorphic coefficients is investigated by the Nevanlinna′s value distribution of meromorphic function,which is generalized into the more general case of complex linear differential equations for composite functions.When more than one coefficient of involved equations have the maximal order or the maximal lower order,some estimates on the lower bound of the order or the lower order of non-zero meromorphic solutions of involved equations are obtained under some conditions.
The growth of meromorphic solutions of a kind of homogenous and non-homogeneous complex linear equations for composite functions with meromorphic coefficients is investigated by the Nevanlinna′s value distribution of meromorphic function,which is generalized into the more general case of complex linear differential equations for composite functions.When more than one coefficient of involved equations have the maximal order or the maximal lower order,some estimates on the lower bound of the order or the lower order of non-zero meromorphic solutions of involved equations are obtained under some conditions.
引文
[1] Hayman W K.Meromorphic functions [M].Oxford:Clarendon Press,1964.
[2] Laine I.Nevanlinna theory and complex differential equations [M].Berlin:Walter de Gruyter,1993.
[3] 仪洪勋,杨重骏.亚纯函数唯一性理论 [M].北京:科学出版社,1995.
[4] 杨乐.值分布论及其新研究 [M].北京:科学出版社,1982.
[5] Bernal L G.On growth k-order of solutions of a complex homogeneous linear differential equation [J].Proc Amer Math Soc,1987,101(2):317-322.
[6] Hu Hui,Zheng Xiumin.Growth of solutions of linear differential equations with meromorphic coefficients of [p,q]-order [J].Math Commun,2014,19(1):29-42.
[7] Chiang Yikman,Feng Shaoji.On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane [J].Ramanujan J,2008,16(1):105-129.
[8] Laine I,Yang Chungchun.Clunie theorems for difference and q-difference polynomials [J].J London Math Soc,2007,76(3):556-566.
[9] Liu Kai,Qi Xiaoguang.Meromorphic solutions of q-shift difference equations [J].Ann Polon Math,2011,101(3):215-225.
[10] 涂金,郑秀敏.q-平移差分多项式和推广了的q-平移差分方程亚纯解的一些性质 [J].数学物理学报,2013,33A(5):951-959.
[11] 王珺,张思奇.一类复差分方程亚纯解的增长性问题 [J].复旦学报:自然科学版,2015,54(3):296-300.
[12] Korhonen R.An extension of Picard′s theorem for meromorphic function of small hyper-order [J].J Math Anal Appl,2009,357(1):244-253.
[13] Chen Zongxuan,Shon K H.On growth of meromorphic solutions for linear difference equations [J].Abstr Appl Anal,2013,2013,Article ID:619296,1-6.
[14] Zheng Xiumin,Tu Jin.Growth of meromorphic solutions of linear difference equations [J].J Math Anal Appl,2011,384(2):349-356.
[15] Wu Shunzhou,Zheng Xiumin.Growth of meromorphic solutions of complex linear differential-difference equations with coefficients having the same order [J].J Math Research Appl,2014,34(6):683-695.
[16] Cao Tingbin,Xu Junfeng,Chen Zongxuan.On the meromorphic solutions of linear differential equations on the complex plane [J].J Math Anal Appl,2010,364(1):130-142.
[2] Laine I.Nevanlinna theory and complex differential equations [M].Berlin:Walter de Gruyter,1993.
[3] 仪洪勋,杨重骏.亚纯函数唯一性理论 [M].北京:科学出版社,1995.
[4] 杨乐.值分布论及其新研究 [M].北京:科学出版社,1982.
[5] Bernal L G.On growth k-order of solutions of a complex homogeneous linear differential equation [J].Proc Amer Math Soc,1987,101(2):317-322.
[6] Hu Hui,Zheng Xiumin.Growth of solutions of linear differential equations with meromorphic coefficients of [p,q]-order [J].Math Commun,2014,19(1):29-42.
[7] Chiang Yikman,Feng Shaoji.On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane [J].Ramanujan J,2008,16(1):105-129.
[8] Laine I,Yang Chungchun.Clunie theorems for difference and q-difference polynomials [J].J London Math Soc,2007,76(3):556-566.
[9] Liu Kai,Qi Xiaoguang.Meromorphic solutions of q-shift difference equations [J].Ann Polon Math,2011,101(3):215-225.
[10] 涂金,郑秀敏.q-平移差分多项式和推广了的q-平移差分方程亚纯解的一些性质 [J].数学物理学报,2013,33A(5):951-959.
[11] 王珺,张思奇.一类复差分方程亚纯解的增长性问题 [J].复旦学报:自然科学版,2015,54(3):296-300.
[12] Korhonen R.An extension of Picard′s theorem for meromorphic function of small hyper-order [J].J Math Anal Appl,2009,357(1):244-253.
[13] Chen Zongxuan,Shon K H.On growth of meromorphic solutions for linear difference equations [J].Abstr Appl Anal,2013,2013,Article ID:619296,1-6.
[14] Zheng Xiumin,Tu Jin.Growth of meromorphic solutions of linear difference equations [J].J Math Anal Appl,2011,384(2):349-356.
[15] Wu Shunzhou,Zheng Xiumin.Growth of meromorphic solutions of complex linear differential-difference equations with coefficients having the same order [J].J Math Research Appl,2014,34(6):683-695.
[16] Cao Tingbin,Xu Junfeng,Chen Zongxuan.On the meromorphic solutions of linear differential equations on the complex plane [J].J Math Anal Appl,2010,364(1):130-142.