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
In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf(z) be a nonconstant entire function of ...In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf(z) be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f(z), f(z+c), and △n cf(z) share 0 CM, then f(z+c)≡Af(z), where A(≠0) is a complex constant. Moreover, let a(z), b(z)( O) ∈ S(f) be periodic entire functions with period c and if f(z) - a(z), f(z + c) - a(z), △cn f(z) - b(z) share 0 CM, then f(z + c) ≡ f(z).展开更多
This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).
In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized order and generalized type of entire functions of slow growth are obta...In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized order and generalized type of entire functions of slow growth are obtained in terms of approximation and interpolation errors.展开更多
Let f be an entire function. A point Zo is called a critical point of f if f′(zo) = O, and f(zo) is called a critical value (or an algebraic singularity) of f. Next a ∈ C is said to be an asymptotic value (or...Let f be an entire function. A point Zo is called a critical point of f if f′(zo) = O, and f(zo) is called a critical value (or an algebraic singularity) of f. Next a ∈ C is said to be an asymptotic value (or a transcendental singularity) of f if there exists a curve Г : [0, 1) → C such that limt→1 F(t) = ∞ and limt→1(f o Г)(t) = a. In this paper we find relations between the asymptotic values of f, 9 and f o 9, relations between critical points of f, 9 and f o 9 and also in the case when the two functions f and 9 are semi-conjugated with another entire function.展开更多
Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy ...Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.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.
In 1958, Baker posed the question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? In order to study this problem of permutable transcendental entire functions, by...In 1958, Baker posed the question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? In order to study this problem of permutable transcendental entire functions, by the properties of permutable transcendental entire functions, we prove that if f and g are permutable transcendental entire functions, then mes (J(f)) = mes (J(g)). Moreover, we give some results about the zero measure of the Julia sets of the permutable transcendental entire functions family.展开更多
There have been lots of papers on the uniqueness theory of entire functions concerning shared-sets in the whole complex plane. However, it seems that the uniqueness theory in an angular domain is not widely investigat...There have been lots of papers on the uniqueness theory of entire functions concerning shared-sets in the whole complex plane. However, it seems that the uniqueness theory in an angular domain is not widely investigated. In this paper, we study the uniqueness of entire functions concerning shared-sets in an angular domain instead of the whole complex plane, and we supply examples to show that Theorem 1 is sharp.展开更多
Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M(r^σ,f)≥M(r,f)σ...Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M(r^σ,f)≥M(r,f)σ+ε.then every component of the Fatou set F(f) is bounded.展开更多
Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a→f′(z) = a, and f′(z) = a →f^(k)(z) = a, then either f = Ce^λz + a or f = Ce^...Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a→f′(z) = a, and f′(z) = a →f^(k)(z) = a, then either f = Ce^λz + a or f = Ce^λz + a(λ - 1)/)λ, where C and ), are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.展开更多
Let f(z) be an entire function of order λ and of finite lower order μ. If the zeros of f(z) accumulate in the vicinity of a finite number of rays, then (a) λ is finite; (b) for every arbitrary number k<...Let f(z) be an entire function of order λ and of finite lower order μ. If the zeros of f(z) accumulate in the vicinity of a finite number of rays, then (a) λ is finite; (b) for every arbitrary number k<sub>1</sub>】1, there exists k<sub>2</sub>】1 such that T(k<sub>1</sub>r, f)≤k<sub>2</sub>T(r, f) for all r≥r<sub>0</sub>. Applying the above results, we prove that if f(z) is extremal for Yang’s inequality p=g/2, then (c) every deficient value of f(z) is also its asymptotic value; (d) every asymptotic value of f(z) is also its deficient value; (e) λ=μ; (f) ∑a≠∞δ5(a, f)≤1-k(μ).展开更多
In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≡ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If fn(z) and(fn(z)) share Q(z) CM, th...In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≡ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If fn(z) and(fn(z)) share Q(z) CM, then f(z) = ce 1 nz, where c is a nonzero constant. This result extends Lv's result from the case of polynomial to small entire function.展开更多
Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , ...Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , our theorems extend the results of (Krivosheev 1991) and (Lelong 1986).展开更多
Given any infinite tree in the plane satisfying certain topological conditions,we construct an entire function f with only two critical values±1 and no asymptotic values such that f-1([-1,1])is ambiently homeomor...Given any infinite tree in the plane satisfying certain topological conditions,we construct an entire function f with only two critical values±1 and no asymptotic values such that f-1([-1,1])is ambiently homeomorphic to the given tree.This can be viewed as a generalization of the result of Grothendieck(see Schneps(1994))to the case of infinite trees.Moreover,a similar idea leads to a new proof of the result of Nevanlinna(1932)and Elfving(1934).展开更多
We study a uniqueness question of entire functions order with their difference operators, and deal with a question in this paper extend the corresponding results obtained by Liu Examples are provided to show that the ...We study a uniqueness question of entire functions order with their difference operators, and deal with a question in this paper extend the corresponding results obtained by Liu Examples are provided to show that the results in this paper, in sharing an entire function of smaller posed by Liu and Yang. The results -Yang and by Liu-Laine respectively. a sense, are the best possible.展开更多
With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and gen...With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and generalize some recent results due to Meng(Math.Bohem.139:89-97,2014).展开更多
In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such t...In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such that if f and its derivative f(k)satisfy E(S,f)= E(S,f(k)),and the zeros of f(z)-d are of multiplicity ≥ k + 1,then f = f(k).展开更多
This paper deals with some uniqueness problems of entire functions concerning differential polynomials that share one value with finite weight in a different form. We obtain some theorems which generalize some results...This paper deals with some uniqueness problems of entire functions concerning differential polynomials that share one value with finite weight in a different form. We obtain some theorems which generalize some results given by Banerjee, Fang and Hua, Zhang and Lin, Zhang, etc.展开更多
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].
基金supported by the Natural Science Foundation of Guangdong Province in China(2014A030313422,2016A030310106,2016A030313745)
文摘In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf(z) be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f(z), f(z+c), and △n cf(z) share 0 CM, then f(z+c)≡Af(z), where A(≠0) is a complex constant. Moreover, let a(z), b(z)( O) ∈ S(f) be periodic entire functions with period c and if f(z) - a(z), f(z + c) - a(z), △cn f(z) - b(z) share 0 CM, then f(z + c) ≡ f(z).
文摘This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).
文摘In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized order and generalized type of entire functions of slow growth are obtained in terms of approximation and interpolation errors.
基金This paper is a main talk on the held in Nanjing, P. R. China, July, 2004.
文摘Let f be an entire function. A point Zo is called a critical point of f if f′(zo) = O, and f(zo) is called a critical value (or an algebraic singularity) of f. Next a ∈ C is said to be an asymptotic value (or a transcendental singularity) of f if there exists a curve Г : [0, 1) → C such that limt→1 F(t) = ∞ and limt→1(f o Г)(t) = a. In this paper we find relations between the asymptotic values of f, 9 and f o 9, relations between critical points of f, 9 and f o 9 and also in the case when the two functions f and 9 are semi-conjugated with another entire function.
基金Supported by National Natural Science Foundation of China(1080113410625107)
文摘Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.
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.
文摘In 1958, Baker posed the question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? In order to study this problem of permutable transcendental entire functions, by the properties of permutable transcendental entire functions, we prove that if f and g are permutable transcendental entire functions, then mes (J(f)) = mes (J(g)). Moreover, we give some results about the zero measure of the Julia sets of the permutable transcendental entire functions family.
基金the National Natural Science Foundation of China (10671109)the Research Foundation of Doctor Points of China (20060422049)+1 种基金the JSPS Post Doctoral Fellowship Programthe Fujian Province Natural Science Foundation (2008J0190)
文摘There have been lots of papers on the uniqueness theory of entire functions concerning shared-sets in the whole complex plane. However, it seems that the uniqueness theory in an angular domain is not widely investigated. In this paper, we study the uniqueness of entire functions concerning shared-sets in an angular domain instead of the whole complex plane, and we supply examples to show that Theorem 1 is sharp.
基金Supported by National Natural Science Foundation of China(Grant Nos.11261002 and 11261069)Natural Science Foundation of Yunnan Province of China(Grant No.2012FZ167)Educational Commission of Yunnan Province of China(Grant No.2012Z121)
文摘Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M(r^σ,f)≥M(r,f)σ+ε.then every component of the Fatou set F(f) is bounded.
基金the NNSF of China(Grant No.10471065)the NSF of Education Department of Jiangsu Province(Grant No.04KJD110001)+1 种基金the SRF for ROCS,SEMthe Presidential Foundation of South China Agricultural University
文摘Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f(z) = a→f′(z) = a, and f′(z) = a →f^(k)(z) = a, then either f = Ce^λz + a or f = Ce^λz + a(λ - 1)/)λ, where C and ), are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.
文摘Let f(z) be an entire function of order λ and of finite lower order μ. If the zeros of f(z) accumulate in the vicinity of a finite number of rays, then (a) λ is finite; (b) for every arbitrary number k<sub>1</sub>】1, there exists k<sub>2</sub>】1 such that T(k<sub>1</sub>r, f)≤k<sub>2</sub>T(r, f) for all r≥r<sub>0</sub>. Applying the above results, we prove that if f(z) is extremal for Yang’s inequality p=g/2, then (c) every deficient value of f(z) is also its asymptotic value; (d) every asymptotic value of f(z) is also its deficient value; (e) λ=μ; (f) ∑a≠∞δ5(a, f)≤1-k(μ).
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.2011QNA25)National Natural Science Foundation of China(Grant No.11271179)
文摘In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≡ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If fn(z) and(fn(z)) share Q(z) CM, then f(z) = ce 1 nz, where c is a nonzero constant. This result extends Lv's result from the case of polynomial to small entire function.
基金Supported by NSFC(60174007)and Shanxi Foundation of Science(20031002)
文摘Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , our theorems extend the results of (Krivosheev 1991) and (Lelong 1986).
文摘Given any infinite tree in the plane satisfying certain topological conditions,we construct an entire function f with only two critical values±1 and no asymptotic values such that f-1([-1,1])is ambiently homeomorphic to the given tree.This can be viewed as a generalization of the result of Grothendieck(see Schneps(1994))to the case of infinite trees.Moreover,a similar idea leads to a new proof of the result of Nevanlinna(1932)and Elfving(1934).
基金Supported by National Natural Science Foundation of China(Grant No.11171184)the Natural Science Foundation of Shandong Province,China(Grant No.Z2008A01)
文摘We study a uniqueness question of entire functions order with their difference operators, and deal with a question in this paper extend the corresponding results obtained by Liu Examples are provided to show that the results in this paper, in sharing an entire function of smaller posed by Liu and Yang. The results -Yang and by Liu-Laine respectively. a sense, are the best possible.
文摘With the notion of weakly weighted sharing and relaxed weighted sharing,we investigate the uniqueness problems of certain type of difference polynomials sharing a small function.The results of the paper extend and generalize some recent results due to Meng(Math.Bohem.139:89-97,2014).
基金Supported by the Natural Science Foundation of Anhui Province (Grant No. KJ2010B124)
文摘In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such that if f and its derivative f(k)satisfy E(S,f)= E(S,f(k)),and the zeros of f(z)-d are of multiplicity ≥ k + 1,then f = f(k).
基金Supported by the Youth Foundation of Education Department of Jiangxi Province (Grant Nos.GJJ10050GJJ10223)
文摘This paper deals with some uniqueness problems of entire functions concerning differential polynomials that share one value with finite weight in a different form. We obtain some theorems which generalize some results given by Banerjee, Fang and Hua, Zhang and Lin, Zhang, etc.