This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is dis...This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.展开更多
Let $ \mathcal{F} $ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $ a/b \notin \mathbb{N}\backslash \{ 1\} $ . If for $ f \in \mathcal{F}, f(z) = a \...Let $ \mathcal{F} $ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $ a/b \notin \mathbb{N}\backslash \{ 1\} $ . If for $ f \in \mathcal{F}, f(z) = a \Rightarrow f'(z) = a $ and $ f'(z) = b \Rightarrow f''(z) = b $ , then $ \mathcal{F} $ is normal. We also construct a non-normal family $ \mathcal{F} $ of meromorphic functions in the unit disk Δ={|z|<1} such that for every $ f \in \mathcal{F}, f(z) = m + 1 \Leftrightarrow f'(z) = m + 1 $ and $ f'(z) = 1 \Leftrightarrow f''(z) = 1 $ in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang.展开更多
In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of ...In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.展开更多
Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the...Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.展开更多
In this paper,we continue to discuss the normality concerning omitted holomorphic function and get the following result.Let F be a family of meromorphic functions on a domain D,k≥4 be a positive integer,and let a(z)a...In this paper,we continue to discuss the normality concerning omitted holomorphic function and get the following result.Let F be a family of meromorphic functions on a domain D,k≥4 be a positive integer,and let a(z)and b(z)be two holomorphic functions on D,where a(z)■0 and f(z)≠∞ whenever a(z)=0.If for any f∈F,f'(z)?a(z)f^k(z)≠b(z),then F is normal on D.展开更多
k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The e...k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel.Furthermore,the integral representation for the solution is obtained.展开更多
Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h（z） has no common zeros ...Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h（z） has no common zeros with any f∈F. Assume also that the following two conditions hold for every f ∈F ：（a） f（z） = 0 == f （z） = h（z）; and （b） y（z） = h（z） == ｜f（k）（z）｜ ≤ c, where c is a constant.Then F is normal on D.展开更多
Given an admissible weight w and 0<p<∞, the estimate∫ D|f(z)| pw(z)dm(z)～|f(0)| p+∫ D|f′(z)| p ψ p(z)w(z)dm(z)is valid for all holomorphic functions f in the unit disc D. Here,ψ(r)=∫ 1 rw(t)dtw(r...Given an admissible weight w and 0<p<∞, the estimate∫ D|f(z)| pw(z)dm(z)～|f(0)| p+∫ D|f′(z)| p ψ p(z)w(z)dm(z)is valid for all holomorphic functions f in the unit disc D. Here,ψ(r)=∫ 1 rw(t)dtw(r) is the distortion of w. As an application of the above estimate, it is proved that the Cesàro operator C[·] is bounded on the weighted Bergman spaces L p a,w (D).展开更多
In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤...In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤u≤∞, 1≤u, q≤∞. (iii) 1≤v≤2≤q≤∞, and 0<p≤u≤∞or 1≤p, u≤∞. The first case extends the result of Blasco, Jevtic, and Pavlovic in one variable. The third case generalizes partly the results of Jevtic, Jovanovic, and Wojtaszczyk to higher dimensions.展开更多
In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a poly...In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D.展开更多
文摘This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi-linearity is discussed, in addition to the examination of properties and classifications of linear ring spaces. Particularly, the ring of holomorphic functions on a region of the complex plane is examined, and the manner in which it generates an iterated linear ring space under the complex derivative operator. This notion is then generalized to all rings with nth order linear and surjective operators. Basic operator theory regarding the classifications of linear ring maps is also covered.
基金supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094)the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001)the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry
文摘Let $ \mathcal{F} $ be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that $ a/b \notin \mathbb{N}\backslash \{ 1\} $ . If for $ f \in \mathcal{F}, f(z) = a \Rightarrow f'(z) = a $ and $ f'(z) = b \Rightarrow f''(z) = b $ , then $ \mathcal{F} $ is normal. We also construct a non-normal family $ \mathcal{F} $ of meromorphic functions in the unit disk Δ={|z|<1} such that for every $ f \in \mathcal{F}, f(z) = m + 1 \Leftrightarrow f'(z) = m + 1 $ and $ f'(z) = 1 \Leftrightarrow f''(z) = 1 $ in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang.
基金supported by National Natural Science Foundation of China (10871145 10926066)+1 种基金Doctoral Program Foundation of the Ministry of Education of China (20090072110053)Natural Science Foundation of Zhejiang Province (Y6100007)
文摘In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.
文摘Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.
基金Supported by the NNSF of China(Grant Nos.11761069 and 11871216)Young Teacher Scientific Research Foundation of Xinjiang Normal University(XJNU201506)“13th Five-Year” Plan for Key Discipline Mathematics Bidding Project(Grant No.17SDKD1104),Xinjiang Normal University
文摘In this paper,we continue to discuss the normality concerning omitted holomorphic function and get the following result.Let F be a family of meromorphic functions on a domain D,k≥4 be a positive integer,and let a(z)and b(z)be two holomorphic functions on D,where a(z)■0 and f(z)≠∞ whenever a(z)=0.If for any f∈F,f'(z)?a(z)f^k(z)≠b(z),then F is normal on D.
基金the NSF of China(11571089,11871191)the NSF of Henan Province(222300420397)+1 种基金the NSF of Hebei Province(A2022208007)the Key Foundation of Hebei Normal University(L2018Z01)。
文摘k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel.Furthermore,the integral representation for the solution is obtained.
基金The first author is supported by the Gelbart Research Institute for Mathematical Sciences and by National Natural Science Foundation of China （Grant No. 10671067） the second author is supported by the Israel Science Foundation （Grant No. 395107）
文摘Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h（z） has no common zeros with any f∈F. Assume also that the following two conditions hold for every f ∈F ：（a） f（z） = 0 == f （z） = h（z）; and （b） y（z） = h（z） == ｜f（k）（z）｜ ≤ c, where c is a constant.Then F is normal on D.
基金the1 5 1 Projection and the Natural Science Foundation of Zhejiang Province( M1 0 31 0 4 )
文摘Given an admissible weight w and 0<p<∞, the estimate∫ D|f(z)| pw(z)dm(z)～|f(0)| p+∫ D|f′(z)| p ψ p(z)w(z)dm(z)is valid for all holomorphic functions f in the unit disc D. Here,ψ(r)=∫ 1 rw(t)dtw(r) is the distortion of w. As an application of the above estimate, it is proved that the Cesàro operator C[·] is bounded on the weighted Bergman spaces L p a,w (D).
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471134)grants from Specialized Research Fund for the doctoral program of Higher Education(SRFDP20050358052)Program for New Century Excellent Talents in University(NCET-05-0539).
文摘In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤u≤∞, 1≤u, q≤∞. (iii) 1≤v≤2≤q≤∞, and 0<p≤u≤∞or 1≤p, u≤∞. The first case extends the result of Blasco, Jevtic, and Pavlovic in one variable. The third case generalizes partly the results of Jevtic, Jovanovic, and Wojtaszczyk to higher dimensions.
基金Supported by the Scientific Research Starting Foundation for Master and Ph.D.of Honghe University(XSS08012)Supported by Scientific Research Fund of Yunnan Provincial Education Department of China Grant(09C0206)
文摘In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D.
