In this paper,the growth theorem for convex maps on the Banach space is given, this is: ‖f(x)‖≤‖x‖/(1-‖x‖),x∈B the estimate is best possible for Hilbert space.
In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it....In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it. This type of domain on which we discuss is rather general,in the sense that the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it.展开更多
The authors obtain the growth and covering theorems for strongly starlike mappings of order α on bounded starlike circular domains.This kind of domain discussed is rather general,since the domain must be starlike if ...The authors obtain the growth and covering theorems for strongly starlike mappings of order α on bounded starlike circular domains.This kind of domain discussed is rather general,since the domain must be starlike if exists a normalized biholomorphic starlike mapping on it.展开更多
In this article, we firstly introduce a subclass of parabolic starlike mappings writen as PS(β; ρ).Secondly,the growth theorem for PS(β; ρ) on the unit ball in complex Banach space is obtained. Finaly, as the appl...In this article, we firstly introduce a subclass of parabolic starlike mappings writen as PS(β; ρ).Secondly,the growth theorem for PS(β; ρ) on the unit ball in complex Banach space is obtained. Finaly, as the application of the growth theorem of PS(β; ρ), the distortion theorem along a unit direction is also established.展开更多
Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and cov...Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.展开更多
The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain mus...The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the famous growth and covering theorem for univalent functions.This theorem successfully realizes the initial idea of H. Cartan about how to extend geometric function theory from one variable to several complex variables.展开更多
The growth theorem and the 1/2 covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Li...The growth theorem and the 1/2 covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Liu, Gong, Yu, and Wang. The approach is new, which does not appeal to the automorphisms of the domains; and the domains discussed are rather general on which convex mappings can be studied, since the domain may not have a convex mapping if it is not convex.展开更多
Let f（x） be an almost spirallike mapping of type β with order B on unit ball B of complex Banach space X. In this paper, we consider the growth and covering theorems for f（x）, we also prove that the estimation is...Let f（x） be an almost spirallike mapping of type β with order B on unit ball B of complex Banach space X. In this paper, we consider the growth and covering theorems for f（x）, we also prove that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping. This result include some results known.展开更多
In this paper, we consider growth and covering theorem for f（x）, where f（x） is spiallike mapping of type β with order α defined on unit ball B of complex Banach space, and x=0 is zero of order k＋1 for f（x）-x....In this paper, we consider growth and covering theorem for f（x）, where f（x） is spiallike mapping of type β with order α defined on unit ball B of complex Banach space, and x=0 is zero of order k＋1 for f（x）-x. We also dicate that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping. This result include some results known.展开更多
In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almos...In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almost starlike mappings of order α and starlike mappings of order α are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in Cn is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.展开更多
Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi–circle theorem of such spectrum distribution is obtained — the spectrum distributed within an upper semi...Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi–circle theorem of such spectrum distribution is obtained — the spectrum distributed within an upper semi–circle domain with radius R<SUB>0</SUB> and taking origin as its circle center in the complex plane, and the upper bound estimation of growth rate is given simultaneously. It is found that the smaller the horizontal scale of perturbation is, the higher the model top located, the greater the estimated value of upper bound of such growth rate is. Also the increase of vertical shear of wind and latitude have the positive contribution to the increase of such growth rate. Finally when stratified stability decreases, the relative maximal growth rate increases, while the maximal growth rate decreases instead.展开更多
1. In Ref. [1], Carl H. FitzGerald et al. gave the first result about the growth theorem in several complex variables. They proved that if f is a normalized biholomorphic starlike mapping from the unit ball B^n to C^n,
In this paper, we deduce growth and covering theorem for f（x） by the other means,where f（x） is strongly spirallike mapping of type β with order α defined on Unit Ball B of complex Banach space, and still give gr...In this paper, we deduce growth and covering theorem for f（x） by the other means,where f（x） is strongly spirallike mapping of type β with order α defined on Unit Ball B of complex Banach space, and still give growth upper bound and distortion upper bound for subordinate mapping.展开更多
In this paper, we deduce growth and covering theorem for f（x） by the othermeans where f（x） is strongly spiral-like mapping of type β with order α defined on UnitBall B of complex Banach space and still give gr...In this paper, we deduce growth and covering theorem for f（x） by the othermeans where f（x） is strongly spiral-like mapping of type β with order α defined on UnitBall B of complex Banach space and still give growth upper bound and distortionbound for subordinate mapping.展开更多
In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R...In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R^N(N > 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ_1, λ = λ_?(? = 2, 3, · · ·), and λ_? is the eigenvalues of the operator(-?_N, W_0^(1,N)(?)),which is defined by the Z_2-cohomological index.展开更多
For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includ...For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.展开更多
In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain...In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain pass theorem.展开更多
文摘In this paper,the growth theorem for convex maps on the Banach space is given, this is: ‖f(x)‖≤‖x‖/(1-‖x‖),x∈B the estimate is best possible for Hilbert space.
文摘In this paper,we extend the definition of almost starlike mappings of order 1/2 on B n to the almost starlike mappings of order α on the bounded starlike circular domains in C n,and give the growth theorem of it. This type of domain on which we discuss is rather general,in the sense that the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it.
文摘The authors obtain the growth and covering theorems for strongly starlike mappings of order α on bounded starlike circular domains.This kind of domain discussed is rather general,since the domain must be starlike if exists a normalized biholomorphic starlike mapping on it.
基金Supported by the Doctoral Foundation of Pingdingshan University（PXY-BSQD-20150 05） Supported by the Natural Science Foundation of Zhejiang Province（Y14A010047）+1 种基金 Supported by the the Key Scientific Research Projects in Universities of Henan Province（16Bl10010） Supported by the Foster Foundation of Pingdingshan University（PXY-PYJJ2016007）
文摘In this article, we firstly introduce a subclass of parabolic starlike mappings writen as PS(β; ρ).Secondly,the growth theorem for PS(β; ρ) on the unit ball in complex Banach space is obtained. Finaly, as the application of the growth theorem of PS(β; ρ), the distortion theorem along a unit direction is also established.
基金Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science (Nos.19540205,200717540138,2007).
文摘Let B be the unit ball in a complex Banach space. Let S^＊k＋1（B） be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k ＋ 1 of f（z） - z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S^＊k＋1（B）.
文摘The authors obtain the growth and covering theorem for the class of normalized biholomorphic starlike mappings on bounded starlike circular domains.This type of domain discussed is rather general, since the domain must be starlike if there exists a normalized biholomorphic starlike mapping on it. In the unit disc, it is just the famous growth and covering theorem for univalent functions.This theorem successfully realizes the initial idea of H. Cartan about how to extend geometric function theory from one variable to several complex variables.
文摘The growth theorem and the 1/2 covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Liu, Gong, Yu, and Wang. The approach is new, which does not appeal to the automorphisms of the domains; and the domains discussed are rather general on which convex mappings can be studied, since the domain may not have a convex mapping if it is not convex.
基金Supported by the National Natural Science Foundation of China(10271117)
文摘Let f（x） be an almost spirallike mapping of type β with order B on unit ball B of complex Banach space X. In this paper, we consider the growth and covering theorems for f（x）, we also prove that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping. This result include some results known.
基金Foundation item： Supported by the National Natural Science Foundation of china（10571044）
文摘In this paper, we consider growth and covering theorem for f（x）, where f（x） is spiallike mapping of type β with order α defined on unit ball B of complex Banach space, and x=0 is zero of order k＋1 for f（x）-x. We also dicate that the estimation is precise when β=0 and still give growth upper bound and distortion upper bound for subordinate mapping. This result include some results known.
基金The research was supported by the National Nat ural Science Foundation of China(10571164)Specialized Research Fund for the Doctoral Program of Higher Education(20050358052)+1 种基金Guangdong Natural Science Foundation(06301315)the Doctoral Foundation of Zhanjiang Normal University(Z0420)
文摘In this article, a normalized biholomorphic mapping f defined on bounded starlike circular domain in Cn is considered, where z = 0 is a zero of order k ＋ 1 of f（z） - z. The sharp growth, covering theorems for almost starlike mappings of order α and starlike mappings of order α are established. Meanwhile, the construction of the above mappings on bounded starlike circular domain in Cn is also discussed, it provides the extremal mappings for the growth, covering theorems of the above mappings.
基金The initial study on the instability of such perturbation is mainly supported by National Special Key Project Fund (No. G1998040
文摘Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi–circle theorem of such spectrum distribution is obtained — the spectrum distributed within an upper semi–circle domain with radius R<SUB>0</SUB> and taking origin as its circle center in the complex plane, and the upper bound estimation of growth rate is given simultaneously. It is found that the smaller the horizontal scale of perturbation is, the higher the model top located, the greater the estimated value of upper bound of such growth rate is. Also the increase of vertical shear of wind and latitude have the positive contribution to the increase of such growth rate. Finally when stratified stability decreases, the relative maximal growth rate increases, while the maximal growth rate decreases instead.
文摘1. In Ref. [1], Carl H. FitzGerald et al. gave the first result about the growth theorem in several complex variables. They proved that if f is a normalized biholomorphic starlike mapping from the unit ball B^n to C^n,
文摘In this paper, we deduce growth and covering theorem for f（x） by the other means,where f（x） is strongly spirallike mapping of type β with order α defined on Unit Ball B of complex Banach space, and still give growth upper bound and distortion upper bound for subordinate mapping.
基金Supported by the National Natural Science Foundation of China（10571044）
文摘In this paper, we deduce growth and covering theorem for f（x） by the othermeans where f（x） is strongly spiral-like mapping of type β with order α defined on UnitBall B of complex Banach space and still give growth upper bound and distortionbound for subordinate mapping.
基金Supported by Shanghai Natural Science Foundation(15ZR1429500)NNSF of China(11471215)
文摘In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R^N(N > 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ_1, λ = λ_?(? = 2, 3, · · ·), and λ_? is the eigenvalues of the operator(-?_N, W_0^(1,N)(?)),which is defined by the Z_2-cohomological index.
文摘For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.
文摘In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain pass theorem.
