We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X...We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X.Finally we strengthen the known Bonnesen isoperimetric inequality.展开更多
The purpose of this paper is twofold.First,by using the hyperbolic metric,we establish the Bohr radius for analytic functions from shifted disks containing the unit disk D into convex proper domains of the complex pla...The purpose of this paper is twofold.First,by using the hyperbolic metric,we establish the Bohr radius for analytic functions from shifted disks containing the unit disk D into convex proper domains of the complex plane.As a consequence,we generalize the Bohr radius of Evdoridis,Ponnusamy and Rasila based on geometric idea.By introducing an alternative multidimensional Bohr radius,the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball B of a complex Banach space X.Notice that when B is the unit ball of the complex Hilbert space X,we show that the constant 1/3 is the Bohr radius for normalized convex mappings of B,which generalizes the result of convex functions on D.展开更多
In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed an...In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.展开更多
By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X an...By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.展开更多
In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic functio...In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic function. To attain our purpose we first establish a fundamental inequality for the modulus of derivative of a holomorphic covering mapping whose image is an annulus by virtue of the hyperbolic metric. The inequality is of independent significance. We make a simple survey on some domain constants for hyperbolic domains.展开更多
Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ...Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ε=1 as |a_0|≤1 and ε=-1 as |a_0|>1. This result is a precise version of the well-known theorem of Landau and an improvement of the results of W. Lai~[1], J. A. Hempel~[2] and J. A. Jenkins~[3]展开更多
Let f be analytic in a hyperbolic region Ω. The Bloch constant β_f of f is defined by β_f=sup z∈Ω|f’(z)|/λ_Ω(z), where λ_Ω(z)|dz|is the Poincare metric in Ω. Suppose △is hyperbolic and lim inf ω→cλ△(w...Let f be analytic in a hyperbolic region Ω. The Bloch constant β_f of f is defined by β_f=sup z∈Ω|f’(z)|/λ_Ω(z), where λ_Ω(z)|dz|is the Poincare metric in Ω. Suppose △is hyperbolic and lim inf ω→cλ△(w)】λ(△)】0, (?)c∈(?)△ where λ(△)=inf w∈△ λ△(w). Then for all f with f(Ω)(?)△, we have β_f≤1/λ(△). In this paper we study the extremal functions defined by β_f=1/λ(△) and the existence of those functions.展开更多
In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of m...In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.展开更多
In the previous paper (JMP 2014) we showed that there exists a NeoMinkowskian Gravitational Expanding Solution of GR (General Relativity) with CC (Cosmological Constant). We prove now that NeoMinkowskian Vacuum (non-b...In the previous paper (JMP 2014) we showed that there exists a NeoMinkowskian Gravitational Expanding Solution of GR (General Relativity) with CC (Cosmological Constant). We prove now that NeoMinkowskian Vacuum (non-baryonic Fluid), with gravitational (first) density (dark energy) and gravitational waves (at light speed), corresponds to the Gravitation Field of a Cosmological Black Hole (CBH). The latter predicts furthermore a basic emission of Radiation (CBR) from Hubble spherical singular Horizon to the inside of CBH (unlike Hawking’s emission) at an initial singular time. Our solution is then compatible with a well-tempered Big Bang and Expanding Universe (Escher’s Figure, see Penrose, 3) but incompatible with inflation. The latter is based on Hypothesis of a so-called Planck’s particle (Lemaitre’s primitive atom) characterized by a so-called Planck length. We prove that we can short-circuit this unstable particle with a stable cosmological Poincaré’s electron with gravific pressure. It is well known that electron is a stranger in usual Minkowskian vacuum (dixit Einstein). The stranger electron can be perfectly integrated in NeoMinkowskian Radiation fluid and then also (with its mass, charge and wavelength) in (second density of) CBR. Everything happens as if the leptonic mass of the electron were induced by our cosmological field. The unexpected cosmological model proposed here is the only one that predicts numerical values of (second) density and temperature of CBR very close to the observed (COBE) values.展开更多
Let φ be a normal function on [0,1] and A^1(φ)the Bergman space withweight φ(|z|)/(1-|z|~2).An atomic decomposition theorem for A^1(φ)is obtained,and a necessary and nearly-sufficient condition is given to make a ...Let φ be a normal function on [0,1] and A^1(φ)the Bergman space withweight φ(|z|)/(1-|z|~2).An atomic decomposition theorem for A^1(φ)is obtained,and a necessary and nearly-sufficient condition is given to make a sequence of pointsinterpolate for A^1(φ).展开更多
Let λG（z）|dz| be the hyperbolic metric on a simply connected proper domain G?C containing the origin, and let ■be the Banach norms of Cnj for j = 1, 2,…,k.This note is to prove that if f is a normalized biholomor...Let λG（z）|dz| be the hyperbolic metric on a simply connected proper domain G?C containing the origin, and let ■be the Banach norms of Cnj for j = 1, 2,…,k.This note is to prove that if f is a normalized biholomorphic convex function on G, then ■is a normalized biholomorphic convex mapping on ■where N = 1+n1 + … + nk and the branch is chosen such that ■,j = 1,…, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.展开更多
In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles co...In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles concerning relationships among the above mentioned numbers are formulated.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10971167)
文摘We first estimate the containment measure of a convex domain to contain in another in a surface X of constant curvature.Then we obtain the analogue of the known Bonnesen isoperimetric inequality for convex domain in X.Finally we strengthen the known Bonnesen isoperimetric inequality.
基金supported by the National Natural Science Foundation of China(12071161,11971165&11671362)the Natural Science Foundation of Fujian Province(2020J01073)。
文摘The purpose of this paper is twofold.First,by using the hyperbolic metric,we establish the Bohr radius for analytic functions from shifted disks containing the unit disk D into convex proper domains of the complex plane.As a consequence,we generalize the Bohr radius of Evdoridis,Ponnusamy and Rasila based on geometric idea.By introducing an alternative multidimensional Bohr radius,the second purpose is to obtain the Bohr radius of higher dimensions for Carathéodory families in the unit ball B of a complex Banach space X.Notice that when B is the unit ball of the complex Hilbert space X,we show that the constant 1/3 is the Bohr radius for normalized convex mappings of B,which generalizes the result of convex functions on D.
基金The project supported in part by the National Natural Science Foundation of China under Grant No. 10671124 and the Program for New Century Excellent Talents in University of China under Grant No. NCET-05-0390 Acknowledgments The author would like to thank the Center of Mathematical Sciences at Zhejiang University for the great support and hospitality and the referee for pertinent comments and valuable suggestions.
文摘In this paper we investigate the Einstein＇s hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein＇s hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow.
基金Supported by NNSF for Young Scientists of China(Grant No.11101290)NNSF of China(Grant No.11071179)
文摘By explicit constructions,we give direct proofs of the following results： for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 〉1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX（α） = lX（β）,extX（α） = extX（β） and lq（α） = lq（β）,where lX（·）,extX（·） and lq（·） are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.
基金supported by National Natural Science Foundation of China (Grant No.10871108)
文摘In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic function. To attain our purpose we first establish a fundamental inequality for the modulus of derivative of a holomorphic covering mapping whose image is an annulus by virtue of the hyperbolic metric. The inequality is of independent significance. We make a simple survey on some domain constants for hyperbolic domains.
基金Project supported by the National Natural Science Foundation of China
文摘Let f(z)=a_o+a_1z+… be holomorphic in the unit disk {z: |z|<1} and omit the values 0 and 1. It is proved in this paper that |a_1|≤2|a_0|{|log|a_0||+Γ~4(1/4)/4π~2-mRe(a_n^8 + 1)} where m>0.04 is a constant, ε=1 as |a_0|≤1 and ε=-1 as |a_0|>1. This result is a precise version of the well-known theorem of Landau and an improvement of the results of W. Lai~[1], J. A. Hempel~[2] and J. A. Jenkins~[3]
基金Supported by the National Natural Science Foundation of China.
文摘Let f be analytic in a hyperbolic region Ω. The Bloch constant β_f of f is defined by β_f=sup z∈Ω|f’(z)|/λ_Ω(z), where λ_Ω(z)|dz|is the Poincare metric in Ω. Suppose △is hyperbolic and lim inf ω→cλ△(w)】λ(△)】0, (?)c∈(?)△ where λ(△)=inf w∈△ λ△(w). Then for all f with f(Ω)(?)△, we have β_f≤1/λ(△). In this paper we study the extremal functions defined by β_f=1/λ(△) and the existence of those functions.
基金supported by National Natural Science Foundation of China (Grant Nos.10671004,10831004)The Doctoral Education Program Foundation (Grant No.20060001003)
文摘We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.
基金supported by National Natural Science Foundation of China(Grant No.11071074)
文摘In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.
文摘In the previous paper (JMP 2014) we showed that there exists a NeoMinkowskian Gravitational Expanding Solution of GR (General Relativity) with CC (Cosmological Constant). We prove now that NeoMinkowskian Vacuum (non-baryonic Fluid), with gravitational (first) density (dark energy) and gravitational waves (at light speed), corresponds to the Gravitation Field of a Cosmological Black Hole (CBH). The latter predicts furthermore a basic emission of Radiation (CBR) from Hubble spherical singular Horizon to the inside of CBH (unlike Hawking’s emission) at an initial singular time. Our solution is then compatible with a well-tempered Big Bang and Expanding Universe (Escher’s Figure, see Penrose, 3) but incompatible with inflation. The latter is based on Hypothesis of a so-called Planck’s particle (Lemaitre’s primitive atom) characterized by a so-called Planck length. We prove that we can short-circuit this unstable particle with a stable cosmological Poincaré’s electron with gravific pressure. It is well known that electron is a stranger in usual Minkowskian vacuum (dixit Einstein). The stranger electron can be perfectly integrated in NeoMinkowskian Radiation fluid and then also (with its mass, charge and wavelength) in (second density of) CBR. Everything happens as if the leptonic mass of the electron were induced by our cosmological field. The unexpected cosmological model proposed here is the only one that predicts numerical values of (second) density and temperature of CBR very close to the observed (COBE) values.
基金Supported by Doctoral Program Foundation of Institute of Higher Education
文摘Let φ be a normal function on [0,1] and A^1(φ)the Bergman space withweight φ(|z|)/(1-|z|~2).An atomic decomposition theorem for A^1(φ)is obtained,and a necessary and nearly-sufficient condition is given to make a sequence of pointsinterpolate for A^1(φ).
基金partially supported by the National Natural Science Foundation of China(11671362,11571105)Beijing Municipal Natural Science Foundation(1182008)the Scientific Research Funds of Huaqiao University
文摘Let λG（z）|dz| be the hyperbolic metric on a simply connected proper domain G?C containing the origin, and let ■be the Banach norms of Cnj for j = 1, 2,…,k.This note is to prove that if f is a normalized biholomorphic convex function on G, then ■is a normalized biholomorphic convex mapping on ■where N = 1+n1 + … + nk and the branch is chosen such that ■,j = 1,…, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.
基金supported by the National Natural Science Foundation of China (Nos. 11771118,11801336, 12171400)the Innovation Fund Designated for Graduate Students of Hebei Province (No.CXZZBS2018101)+1 种基金China Scholarship Council (CSC for short)China Postdoctoral Science Foundation (No. 2021M691889).
文摘In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles concerning relationships among the above mentioned numbers are formulated.
