关于二阶线性复微分方程解的Borel方向
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摘要
利用亚纯函数的Nevanlinna值分布理论,研究了二阶线性复微分方程f"+A(z)f'+B(z)f=0的解的Borel方向,其中A(z)是满足杨不等式极端情况的整函数.证明了当B(z)满足适当条件时,方程的每一个非平凡解为无穷级,并且计算了方程解的Borel方向的个数.
In this paper, we consider the Borel directions of solutions of the differential equation f" + A(z)f + B(z)f = 0. By using Nevanlinna's value distribution theory and assuming that A(z) is extremal for Yang's inequality, we provide conditions for B(z) that guarantee that every non-trivial solution f of the equation is of infinite order; we also calculate the number of Borel directions of these solutions.
In this paper, we consider the Borel directions of solutions of the differential equation f" + A(z)f + B(z)f = 0. By using Nevanlinna's value distribution theory and assuming that A(z) is extremal for Yang's inequality, we provide conditions for B(z) that guarantee that every non-trivial solution f of the equation is of infinite order; we also calculate the number of Borel directions of these solutions.
引文
[1] HAYMAN W. Meromorphic Function[M]. Oxford:Clarendon Press, 1964.
[2]杨乐.值分布论及其新研究[M].北京:科学出版社,1982.
[3]WU S J. On the location of zeros of solutions of f"+Af=0 where A(z)is entire[J]. Math Scand, 1994, 74(2):293-312.
[4]LAINE I, WU S J. Removable sets in the oscillation theory of complex differential equations[J]. Math Anal Appl, 1997, 214(1):233-244.
[5]GOLDBERG A A, OSTROVSKII I V. Distribution of Values of Meromorphic Functions[M]. Moscow:Izdat Nauk, 1970.
[6]蒋业阳,陈宗煊.非齐次线性微分方程解的增长性[J].数学年刊,2013, 34(3):291-298.
[7] LONG J R, WU P C, ZHANG Z. On the growth of solutions of second order linear differential equations with extremal coefficients[J]. Acta Mathematica Sinica(English Ser), 2013, 29(2):365-372.
[8] WU S J. Some results on entire functions of finite lower order[J]. Acta Math Sinica(English Ser), 1994, 10(2):168-178.
[9] HUANG Z G, WANG J. On the radial distribution of Julia sets of entire solutions of f~((n))A(z)f=0[J]. Math Anal Appl, 2012, 387(2):1106-1113.
[10]易才凤,钟文波.二阶微分方程f"+Af'+Bf=0解的增长性[J].江西师范大学学报(自然科学版).2015, 39(4):340-344.
[11] ZHENG J H. Value Dstribution of Meromorphic Functions[M]. Beijing:Tsinghua Univ Press, 2010.
[12] ZHENG J H. On multiply-connected Fatou components in iteration of meromorphic functions[J]. Math Anal Appl, 2006, 313(1):24-37.
[2]杨乐.值分布论及其新研究[M].北京:科学出版社,1982.
[3]WU S J. On the location of zeros of solutions of f"+Af=0 where A(z)is entire[J]. Math Scand, 1994, 74(2):293-312.
[4]LAINE I, WU S J. Removable sets in the oscillation theory of complex differential equations[J]. Math Anal Appl, 1997, 214(1):233-244.
[5]GOLDBERG A A, OSTROVSKII I V. Distribution of Values of Meromorphic Functions[M]. Moscow:Izdat Nauk, 1970.
[6]蒋业阳,陈宗煊.非齐次线性微分方程解的增长性[J].数学年刊,2013, 34(3):291-298.
[7] LONG J R, WU P C, ZHANG Z. On the growth of solutions of second order linear differential equations with extremal coefficients[J]. Acta Mathematica Sinica(English Ser), 2013, 29(2):365-372.
[8] WU S J. Some results on entire functions of finite lower order[J]. Acta Math Sinica(English Ser), 1994, 10(2):168-178.
[9] HUANG Z G, WANG J. On the radial distribution of Julia sets of entire solutions of f~((n))A(z)f=0[J]. Math Anal Appl, 2012, 387(2):1106-1113.
[10]易才凤,钟文波.二阶微分方程f"+Af'+Bf=0解的增长性[J].江西师范大学学报(自然科学版).2015, 39(4):340-344.
[11] ZHENG J H. Value Dstribution of Meromorphic Functions[M]. Beijing:Tsinghua Univ Press, 2010.
[12] ZHENG J H. On multiply-connected Fatou components in iteration of meromorphic functions[J]. Math Anal Appl, 2006, 313(1):24-37.