Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three d...Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.展开更多
In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′j...In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′js are non-zero constants.We obtain the expressions of meromorphic solutions of the above equations under some restrictions onα′js.Some examples are given to illustrate the possibilities of our results.展开更多
In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain ...In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain the necessary conditions on the degree of R(z,f)for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of Q(z,f)on f.Finally,we provide some examples to illustrate that all cases occur.展开更多
The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic...The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic solutions of hyper-order strictly less than 1 to the Fermat-type functional equation(a0f(z)+a1f(z+c))^(3)+(b0f(z)+b1f(z+c))3=e^(αz+β),where a0,a1,b0,b1,α,β,c are complex constants and c≠0.展开更多
This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities...This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.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
Let f and g be two nonconstant meromorphic functions,and n be a positive integer.If f^(n)(z)and g^(n)(z)share 1 CM(counting multiplicities),f(z)and g(z)share∞IM(ignoring multiplicities),and N(r,1f)+N(r,1g)=S(r,f)+S(r...Let f and g be two nonconstant meromorphic functions,and n be a positive integer.If f^(n)(z)and g^(n)(z)share 1 CM(counting multiplicities),f(z)and g(z)share∞IM(ignoring multiplicities),and N(r,1f)+N(r,1g)=S(r,f)+S(r,g),then either f(z)≡tg(z)or f(z)g(z)≡t,where t^(n)=1.As an application,shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.展开更多
In this paper,we consider the truncated multiplicity finite range set problem of meromorphic functions on some complex disc.By using the value distribution theory of meromorphic functions,we establish a second main th...In this paper,we consider the truncated multiplicity finite range set problem of meromorphic functions on some complex disc.By using the value distribution theory of meromorphic functions,we establish a second main theorem for meromorphic functions with finite growth index which share meromorphic functions(may not be small functions).As its application,we also extend the result of a finite range set with truncated multiplicity.展开更多
Aim To study the value distribution of meromorphic functions in angular domains, the deficiency, the deficient value, the Nevanlinna direction and other singular directions. Methods A fundamental inequality of Nevan...Aim To study the value distribution of meromorphic functions in angular domains, the deficiency, the deficient value, the Nevanlinna direction and other singular directions. Methods A fundamental inequality of Nevanlinna characteristic functions in the angular domain was used, which is similar with the Nevanlinna secondary fundamental theorem. Results The deficiency and deficient value of meromorphic functions about an angular domain and a direction were defined. The definition of Nevanlinna direction was improved. Conclusion For a family of meromorphic functions, it is proved that the number of deficient values is at most countable and the sum of deficiencies isnt greater than 2. The existence of the Nevanlinna direction is obtained. The existence of Borel and Julia directions and the relation between them are found.展开更多
The uniqueness of meromorphic functions with one sharing value and an equality on deficiency is studied. We show that if two nonconstant meromorphic functions f(z) and g(z) satisfy δ(0,f)+δ(0,g)+δ(∞,f)+δ(∞,g)=3 ...The uniqueness of meromorphic functions with one sharing value and an equality on deficiency is studied. We show that if two nonconstant meromorphic functions f(z) and g(z) satisfy δ(0,f)+δ(0,g)+δ(∞,f)+δ(∞,g)=3 or δ 2(0,f)+δ 2(0,g)+δ 2(∞,f)+δ 2(∞,g)=3, and E(1,f)=E(1,g) then f(z),g(z) must be one of five cases.展开更多
This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect...This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.展开更多
In this paper, we investigate the bounds of the coefficients of several classes of meromorphic bi-univalent functions. The results presented in this paper improve or generalize the recent works of other authors.
In this paper, by using the normality criteria for K quasimeromorphic mapping of several complex variables, we get a normality criteria for families of holomorphic functions and of meromorphic functions family.
We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex dif...We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.展开更多
In this article, we mainly investigate the growth and existence of meromorphic solutions of a type of systems of composite functional equations, and obtain some interesting results. It extends some results concerning ...In this article, we mainly investigate the growth and existence of meromorphic solutions of a type of systems of composite functional equations, and obtain some interesting results. It extends some results concerning functional equations to the systems of functional equations.展开更多
In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based o...In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.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 this article, we deal with the uniqueness problems on meromorphic functions sharing two finite sets in an angular domain instead of the whole plane C. In particular, we investigate the uniqueness for meromorphic fu...In this article, we deal with the uniqueness problems on meromorphic functions sharing two finite sets in an angular domain instead of the whole plane C. In particular, we investigate the uniqueness for meromorphic functions of infinite order in an angular domain and obtain some results. Moreover, examples show that the conditions in theorems are necessary.展开更多
Let S1 = {∞} and S2 = {ω ： Ps（ω） = 0}, Ps（ω） being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconsta...Let S1 = {∞} and S2 = {ω ： Ps（ω） = 0}, Ps（ω） being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f^-1 （Si） = g^-1 （Si） （i = 1, 2）, where f^-1 （Si） and g^-1 （Si） denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.展开更多
We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.
基金Supported by the National Natural Science Foundation of China (Grant No.12161074)the Talent Introduction Research Foundation of Suqian University (Grant No.106-CK00042/028)+1 种基金Suqian Sci&Tech Program (Grant No.M202206)Sponsored by Qing Lan Project of Jiangsu Province and Suqian Talent Xiongying Plan of Suqian。
文摘Throughout this work,we explore the uniqueness properties of meromorphic functions concerning their interactions with complex differential-difference polynomial.Under the condition of finite order,we establish three distinct uniqueness results for a meromorphic function f associated with the differential-difference polynomial L_(η)^(n)f=Σ_(k=0)^(n)a_(k)f (z+k_(η))+a_(-1)f′.These results lead to a refined characterization of f (z)≡L_(η)^(n)f (z).Several illustrative examples are provided to demonstrate the sharpness and precision of the results obtained in this study.
基金supported by the National Natural Science Foundation of China(No.12001117)the Guangdong Basic and Applied Basic Research Foundation(No.2021A1515110654).
文摘In this article,we study the meromorphic solutions of the following non-linear differential equation■where n and k are integers with n≥k≥3,P_(d)(z,f)is a differential polynomial in f of degree d≤n−1,p′js andα′js are non-zero constants.We obtain the expressions of meromorphic solutions of the above equations under some restrictions onα′js.Some examples are given to illustrate the possibilities of our results.
基金supported by the National Natural Science Foundation of China(12201420,12231013,11701382 and 11971288)the Guangdong Basic and Applied Basic Research Foundation,China(2021A1515010054)and NCAMS.
文摘In this paper,we focus on the admissible transcendental meromorphic solutions of the following delay Schwarzian differential equations with rational coefficients f(z+1)-f(z-1)+a(z)S(f,z)=P(z,f(z))/Q(z,f(z)).We obtain the necessary conditions on the degree of R(z,f)for these delay differential equations and give a classification of the delay Schwarzian differential equations according to the multiplicities of the root of Q(z,f)on f.Finally,we provide some examples to illustrate that all cases occur.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344)。
文摘The functional equation f(z)^n+g(z)^n=1 can be interpreted as the Fermat-type equations over function field.In this paper,by using Nevanlinna theory of meromorphic functions,we investigate the existence of meromorphic solutions of hyper-order strictly less than 1 to the Fermat-type functional equation(a0f(z)+a1f(z+c))^(3)+(b0f(z)+b1f(z+c))3=e^(αz+β),where a0,a1,b0,b1,α,β,c are complex constants and c≠0.
基金partially supported by the NSFC(12061042)the NSF of Jiangxi(20202BAB201003)+3 种基金the support of the National Science Center(Poland)via grant 2017/25/B/ST1/00931partially supported by the Project PID2021-124472NB-I00funded by MCIN/AEI/10.13039/501100011033by"EFDF A way of making Europe"。
文摘This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.
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.
基金Supported by the National Natural Science Foundation of China(11971344)。
文摘Let f and g be two nonconstant meromorphic functions,and n be a positive integer.If f^(n)(z)and g^(n)(z)share 1 CM(counting multiplicities),f(z)and g(z)share∞IM(ignoring multiplicities),and N(r,1f)+N(r,1g)=S(r,f)+S(r,g),then either f(z)≡tg(z)or f(z)g(z)≡t,where t^(n)=1.As an application,shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.
基金Supported by National Natural Science Foundation of China(12061041)Jiangxi Provincial Natural Science Foundation(20232BAB201003).
文摘In this paper,we consider the truncated multiplicity finite range set problem of meromorphic functions on some complex disc.By using the value distribution theory of meromorphic functions,we establish a second main theorem for meromorphic functions with finite growth index which share meromorphic functions(may not be small functions).As its application,we also extend the result of a finite range set with truncated multiplicity.
文摘Aim To study the value distribution of meromorphic functions in angular domains, the deficiency, the deficient value, the Nevanlinna direction and other singular directions. Methods A fundamental inequality of Nevanlinna characteristic functions in the angular domain was used, which is similar with the Nevanlinna secondary fundamental theorem. Results The deficiency and deficient value of meromorphic functions about an angular domain and a direction were defined. The definition of Nevanlinna direction was improved. Conclusion For a family of meromorphic functions, it is proved that the number of deficient values is at most countable and the sum of deficiencies isnt greater than 2. The existence of the Nevanlinna direction is obtained. The existence of Borel and Julia directions and the relation between them are found.
文摘The uniqueness of meromorphic functions with one sharing value and an equality on deficiency is studied. We show that if two nonconstant meromorphic functions f(z) and g(z) satisfy δ(0,f)+δ(0,g)+δ(∞,f)+δ(∞,g)=3 or δ 2(0,f)+δ 2(0,g)+δ 2(∞,f)+δ 2(∞,g)=3, and E(1,f)=E(1,g) then f(z),g(z) must be one of five cases.
基金supported by the National Natural Science Foundation of China (11101096)
文摘This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.
基金Supported by the Natural Science Foundation of Anhui Provincial Department of Education(Grant Nos.KJ2018-A0833 KJ2018A0839)
文摘In this paper, we investigate the bounds of the coefficients of several classes of meromorphic bi-univalent functions. The results presented in this paper improve or generalize the recent works of other authors.
文摘In this paper, by using the normality criteria for K quasimeromorphic mapping of several complex variables, we get a normality criteria for families of holomorphic functions and of meromorphic functions family.
基金Project Supported by the Natural Science Foundation of China (10471065)the Natural Science Foundation of Guangdong Province (04010474)
文摘We apply Nevanlinna theory of the value distribution of meromorphic functions to study the properties of Nevanlinna counting function and proximity function of meromorphic solutions of a type of systems of complex difference equations. Our results can give estimates on the proximity function and the counting function of solutions of systems of difference equations. This implies that solutions have a relatively large number of poles. It extend some result concerning difference equations to the systems of difference equations.
基金Project supported by NSF of China (10471065)the Natural Science Foundation of Guangdong Province (04010474)
文摘In this article, we mainly investigate the growth and existence of meromorphic solutions of a type of systems of composite functional equations, and obtain some interesting results. It extends some results concerning functional equations to the systems of functional equations.
文摘In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.
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 NNSFC (10671109)the NSFFC(2008J0190)+1 种基金the Research Fund for Talent Introduction of Ningde Teachers College (2009Y019)the Scitific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry
文摘In this article, we deal with the uniqueness problems on meromorphic functions sharing two finite sets in an angular domain instead of the whole plane C. In particular, we investigate the uniqueness for meromorphic functions of infinite order in an angular domain and obtain some results. Moreover, examples show that the conditions in theorems are necessary.
基金This work was supported by the National Natural Science Foundation of China (10671109)Fujian Province Youth Science Technology Program(2003J006)the Doctoral Programme Foundation of Higher Education(20060422049)
文摘Let S1 = {∞} and S2 = {ω ： Ps（ω） = 0}, Ps（ω） being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f^-1 （Si） = g^-1 （Si） （i = 1, 2）, where f^-1 （Si） and g^-1 （Si） denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.
基金supported by Nature Science Foundation of China(11461070),supported by Nature Science Foundation of China(11271227)PCSIRT(IRT1264)
文摘We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.
