Let f（z） be a meromorphic function and ψ be the differential polynomial of f which satisfies the condition of -↑N（r, f）＋-↑N （r, 1/f） = S（r, f）. We obtain several results about the zero point of the ψ and ...Let f（z） be a meromorphic function and ψ be the differential polynomial of f which satisfies the condition of -↑N（r, f）＋-↑N （r, 1/f） = S（r, f）. We obtain several results about the zero point of the ψ and those results extend and improve the results of Yang and Yi in this paper.展开更多
We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H...We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].展开更多
This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extend and improve on some theorems given in [3].
This paper deals with problems of the uniqueness of entire functions that share one pair of values with their derivatives. The results in this paper generalize and improve a result of Jank, Mues and Volkmann, a result...This paper deals with problems of the uniqueness of entire functions that share one pair of values with their derivatives. The results in this paper generalize and improve a result of Jank, Mues and Volkmann, a result of YANG L Z and a result of R Brück.展开更多
In this paper, we deal with the uniqueness problems on entire and meromorphic functions con- cerning differential polynomials that share fixed-points. Moreover, we generalise and improve some results of Weichuan Lin, ...In this paper, we deal with the uniqueness problems on entire and meromorphic functions con- cerning differential polynomials that share fixed-points. Moreover, we generalise and improve some results of Weichuan Lin, Hongxun Yi, Meng Chao, C. Y. Fang, M. L. Fang and Junfeng xu.展开更多
Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, a...Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.展开更多
In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplic...In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.展开更多
We prove an oscillation theorem of two meromorphic functions whose derivatives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10...We prove an oscillation theorem of two meromorphic functions whose derivatives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10], and supplement results given by Nevanlinna [9] and Gundersen [3, 4]. Some examples are provided to show that the results in this paper are best possible.展开更多
In this paper,we deal with the uniqueness problems on entire functions concerning differential polynomials that share one small function.Moreover,we improve some former results of M Fang and W Lin.
The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A sim...The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A simple andelementary way to study such subjects is exhibited in this paper; that is,once an estimation of B(r,*) in terms of a few C(r,**) in the Nevanlinna theory on angular domains is established, we can produce one result that the order of a mermorphic function with radially distributed values related to C(r,**) can be estimated under the assumption of existence of suitable deficient value. The results obtained in this paper lead us to a new singular direction in terms of Nevanlinna characterstic instead of the order of meromorphic functions.展开更多
We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose dif...We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.展开更多
Value distribution theory is concerned with the position and frequency of solutions of the equation f(z) = a. Here f may be entire, i.e. an everywhere convergent power series or meromorphic, i.e. the ratio of two such...Value distribution theory is concerned with the position and frequency of solutions of the equation f(z) = a. Here f may be entire, i.e. an everywhere convergent power series or meromorphic, i.e. the ratio of two such series, or a function in some other domains, such as an angle or a disk. Yang Lo’s significant contributions to this area will be highlighted. Some of his important contributions to normal families will also be described.展开更多
In this note it is proved that entire functions of the form Fa（z）=H（z）-aα（z）where H andα are entire functions, with α having at least one zero, H＇ and α＇having no common zeros and T（r, α） = S（r, H）, a...In this note it is proved that entire functions of the form Fa（z）=H（z）-aα（z）where H andα are entire functions, with α having at least one zero, H＇ and α＇having no common zeros and T（r, α） = S（r, H）, are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.展开更多
In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c...In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c（z）.展开更多
We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Co...We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.展开更多
Value distributions of the general differential monomials is discussed.The following theorem is obtained:Let f be a transcendental meromorphic function in the plane,F=f n 0 (f (i) ) n i …(f (k) ) ...Value distributions of the general differential monomials is discussed.The following theorem is obtained:Let f be a transcendental meromorphic function in the plane,F=f n 0 (f (i) ) n i …(f (k) ) n k -c,n i≥1,c≠0 be a constant then (n 0-2)T(r,f)≤(r,1F)+S(r,f) when n 0】2;T(r,f)≤7(i+1)i( i) (r,1f)+(r,1F))+S(r,f) when n 0=1;T(r,f)≤7(N(r,1f)+(r,1F))+S(r,f) when n 0=0.展开更多
By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n...By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.展开更多
基金Supported by the Natural Science Fundation of Henan Proivince(0211050200)
文摘Let f（z） be a meromorphic function and ψ be the differential polynomial of f which satisfies the condition of -↑N（r, f）＋-↑N （r, 1/f） = S（r, f）. We obtain several results about the zero point of the ψ and those results extend and improve the results of Yang and Yi in this paper.
基金supported by NSF of Fujian Province,China(S0750013),supported by NSF of Fujian Province,China(2008J0190)the Research Foundation of Ningde Normal University(2008J001)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].
文摘This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extend and improve on some theorems given in [3].
文摘This paper deals with problems of the uniqueness of entire functions that share one pair of values with their derivatives. The results in this paper generalize and improve a result of Jank, Mues and Volkmann, a result of YANG L Z and a result of R Brück.
文摘In this paper, we deal with the uniqueness problems on entire and meromorphic functions con- cerning differential polynomials that share fixed-points. Moreover, we generalise and improve some results of Weichuan Lin, Hongxun Yi, Meng Chao, C. Y. Fang, M. L. Fang and Junfeng xu.
基金Supported by the NNSF of China(11071083)the Tianyuan Foundation(11126267)
文摘Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.
基金Supported by the NSF of China(10771220)Supported by the Doctorial Point Fund of National Education Ministry of China(200810780002)
文摘In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.
基金supported by the NSFC (11171184), the NSFC(10771121),the NSFC (40776006) the NSFC & RFBR (Joint Project) (10911120056),the NSF of Shandong Province, China (Z2008A01), and the NSF of Shandong Province, China (ZR2009AM008)
文摘We prove an oscillation theorem of two meromorphic functions whose derivatives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10], and supplement results given by Nevanlinna [9] and Gundersen [3, 4]. Some examples are provided to show that the results in this paper are best possible.
文摘In this paper,we deal with the uniqueness problems on entire functions concerning differential polynomials that share one small function.Moreover,we improve some former results of M Fang and W Lin.
基金This work was supported by the National Natural Science Foundation of China(Grant No.19971049)BRF of Tsinghua University.
文摘The growth of transcendental meromorphic functions in terms of their orders is investigated in this paper when they and their derivatives have radially distributed values following the discussion of the author . A simple andelementary way to study such subjects is exhibited in this paper; that is,once an estimation of B(r,*) in terms of a few C(r,**) in the Nevanlinna theory on angular domains is established, we can produce one result that the order of a mermorphic function with radially distributed values related to C(r,**) can be estimated under the assumption of existence of suitable deficient value. The results obtained in this paper lead us to a new singular direction in terms of Nevanlinna characterstic instead of the order of meromorphic functions.
基金Acknowledgements This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the first and third authors worked as visiting scholars. The authors wish to thank the anonymous referees for their very helpful comments and useful suggestions. This work was also supported by the National Natural Science Foundation of China （Grant No. 11271090）, the Tianyuan Youth Fund of the National Natural Science Foundation of China （Grant No. 11326083）, the Shanghai University Young Teacher Training Program （ZZSDJ12020）, the Innovation Program of Shanghai Municipal Education Commission （14YZ164）, the Natural Science Foundation of Guangdong Province （S2012010010121）, and the Projects （13XKJC01） from the Leading Academic Discipline Project of Shanghai Dianji University.
文摘We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.
基金supported by National Natural Science Foundation of China (Grant Nos. 10671004, 10831004)the Doctoral Education Program Foundation of China (Grant No.20060001003)
文摘Value distribution theory is concerned with the position and frequency of solutions of the equation f(z) = a. Here f may be entire, i.e. an everywhere convergent power series or meromorphic, i.e. the ratio of two such series, or a function in some other domains, such as an angle or a disk. Yang Lo’s significant contributions to this area will be highlighted. Some of his important contributions to normal families will also be described.
基金supported by the Academy of Finland (Grant Nos.210245 and 124954)
文摘In this note it is proved that entire functions of the form Fa（z）=H（z）-aα（z）where H andα are entire functions, with α having at least one zero, H＇ and α＇having no common zeros and T（r, α） = S（r, H）, are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.
基金The NSF (06C417) of Hunan Provincethe QNF (04QN10) of Hunan AgricultureUniversity
文摘In this paper we derive the fundamental inequality of the theorem of meromorphic functions. It extends some results of Yi Hong-xun et al. As one of its application, we then study the value distribution of f^（k）f^n-c（z）.
基金supported by the NSFC(12261044)the STP of Education Department of Jiangxi Province of China(GJJ210302)。
文摘We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.
文摘Value distributions of the general differential monomials is discussed.The following theorem is obtained:Let f be a transcendental meromorphic function in the plane,F=f n 0 (f (i) ) n i …(f (k) ) n k -c,n i≥1,c≠0 be a constant then (n 0-2)T(r,f)≤(r,1F)+S(r,f) when n 0】2;T(r,f)≤7(i+1)i( i) (r,1f)+(r,1F))+S(r,f) when n 0=1;T(r,f)≤7(N(r,1f)+(r,1F))+S(r,f) when n 0=0.
基金Supported by the Nature Science foundation of Henan Province(0211050200)
文摘By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.
