Let f（z） be a transcendental meromorphic function in the complex plane and a ≠0 be a constant, for any positive integer m, n, k, satisfy m ≥ nk＋n＋2, ψ= f^m ＋a（f^（κ））^n has infinitely many zeros. The corre...Let f（z） be a transcendental meromorphic function in the complex plane and a ≠0 be a constant, for any positive integer m, n, k, satisfy m ≥ nk＋n＋2, ψ= f^m ＋a（f^（κ））^n has infinitely many zeros. The corresponding normal criterion also is proved.展开更多
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
In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one...In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.展开更多
The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., ...The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.展开更多
In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and ...In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and g^n＋ ag^（k）share b CM and the b-points of f^n＋ af^（k）are not the zeros of f and g, then f and g are either equal or closely related.展开更多
This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu’s criterion for the normality of a family of meromorphic functions.
Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functi...Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that φ_(k)[f](z)≠0 and|Res(f,a)-j|≥δ for all j∈{0,1,...,k-1}and all simple poles α of f in D.Then F is quasi-normal on D of order 1.展开更多
In this paper, we obtain the following normal criterion： Let F be a family of mero-morphic functions in domain D belong to C, all of whose zeros have multiplicity k ＋ 1 at least. If there exist holomorphic functio...In this paper, we obtain the following normal criterion： Let F be a family of mero-morphic functions in domain D belong to C, all of whose zeros have multiplicity k ＋ 1 at least. If there exist holomorphic functions α（z） not vanishing on D, such that for every function f（z） ∈F, f（z） shares α（z） IM with L（f） on D, then F is normal on D, where L（f） is linear differential polynomials of f（z） with holomorphic coefficients, and k is some positive numbers. We also proved coressponding results on normal functions.展开更多
In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ...In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ∈ N. Moreover, an analogous normality criterion isobtained.展开更多
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit...In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.展开更多
In this paper,we study the problem on the fixed points of the lth power of linear differential polynomials generated by second order linear differential equations.Because of the control of differential equation,we can...In this paper,we study the problem on the fixed points of the lth power of linear differential polynomials generated by second order linear differential equations.Because of the control of differential equation,we can obtain some precise estimate of their fixed points.展开更多
This paper is devoted to studying the relationship between meromorphic functions f（z） and g（z） when their differential polynomials satisfy sharing condition weaker than sharing one value IM.
In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential pol...In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.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
Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex nu...Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f（z）f（k） （z）.展开更多
This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of...This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).展开更多
This paper deals with a class of <em>n</em>-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (<em>n</em> is an odd...This paper deals with a class of <em>n</em>-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (<em>n</em> is an odd number) or two (<em>n</em> is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.展开更多
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&...Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +展开更多
Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers s...Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.展开更多
基金Supported by the NSF of China(10771121)Supported by the "Yumiao" Project of Guangdong Province(LYM08097)
文摘Let f（z） be a transcendental meromorphic function in the complex plane and a ≠0 be a constant, for any positive integer m, n, k, satisfy m ≥ nk＋n＋2, ψ= f^m ＋a（f^（κ））^n has infinitely many zeros. The corresponding normal criterion also is proved.
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The NSF(11301076)of Chinathe NSF(2014J01004)of Fujian Province
文摘In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.
文摘The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.
基金supported by the NNSF(11201014,11171013,11126036,11371225)the YWF-14-SXXY-008,YWF-ZY-302854 of Beihang Universitysupported by the youth talent program of Beijing(29201443)
文摘In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and g^n＋ ag^（k）share b CM and the b-points of f^n＋ af^（k）are not the zeros of f and g, then f and g are either equal or closely related.
文摘This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu’s criterion for the normality of a family of meromorphic functions.
文摘Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that φ_(k)[f](z)≠0 and|Res(f,a)-j|≥δ for all j∈{0,1,...,k-1}and all simple poles α of f in D.Then F is quasi-normal on D of order 1.
基金the"11.5"Research & Study Programe of SWUST(No.06zx2116)
文摘In this paper, we obtain the following normal criterion： Let F be a family of mero-morphic functions in domain D belong to C, all of whose zeros have multiplicity k ＋ 1 at least. If there exist holomorphic functions α（z） not vanishing on D, such that for every function f（z） ∈F, f（z） shares α（z） IM with L（f） on D, then F is normal on D, where L（f） is linear differential polynomials of f（z） with holomorphic coefficients, and k is some positive numbers. We also proved coressponding results on normal functions.
基金Supported by National Natural Science FoundationScience Technology Promotion Foundation of Fujian Province(2003)
文摘In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ∈ N. Moreover, an analogous normality criterion isobtained.
文摘In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.
基金This research is supported by the National Natural Science Foundation of China(No. 10371065) the Natural Science Foundation of Shandong Province, China(No. Z2002A01).
文摘In this paper,we study the problem on the fixed points of the lth power of linear differential polynomials generated by second order linear differential equations.Because of the control of differential equation,we can obtain some precise estimate of their fixed points.
基金Supported by the National Natural Science Foundation of China (Grant Nos.10871047J073010311001057)
文摘This paper is devoted to studying the relationship between meromorphic functions f（z） and g（z） when their differential polynomials satisfy sharing condition weaker than sharing one value IM.
文摘In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.
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Foundation item： Supported by the National Natural Science Foundation of Education Department of Sichuan Province（2002A031） Supported by the ＂11.5＂ Research and Study Programs of SWUST（06zx2116） Supported by the National Natural Science Foundation of China（10271122）
文摘Suppose that function f（z） is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f（z）f（k）（z） takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f（z）f（k） （z）.
基金The second named author was supported in part by an NSERC Postdoctoral Fellowship,Canada and a CR F Grant,University of Alberta
文摘This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).
文摘This paper deals with a class of <em>n</em>-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (<em>n</em> is an odd number) or two (<em>n</em> is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.
文摘Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +
基金Supported by Grant NNSF of China(Grant No.12171491)the Ministerio de Ciencia,Innovación y Universidades,Agencia Estatal de Investigación grants MTM2016-77278-P(FEDER)and PID2019-104658GB-I00(FEDER)+1 种基金the Agència de Gestiód’Ajuts Universitaris i de Recerca grant 2017SGR1617the H2020 European Research Council grant MSCA-RISE-2017-777911。
文摘Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.
