Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>*

In this paper, the authors establish distortion theorems for various subfamilies Hk(B) of holomorphic mappings defined in the unit ball in Cn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to oo. These distortion theorems give lower bound son det f(z) and Redet f'(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies βk (M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When B is the unit disk in C, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f'(z) for locally biholomorphic mappings is also obtained.     

单位球Bn上的Bohr不等式

Bohr＇s type inequalities are studied in this paper： if f is a holomorphic mapping from the unit ball B^n to B^n, f（0）=p, then we have sum from k=0 to∞｜DφP（P）[D^kf（0）（z^k）]｜/k!｜｜DφP（P）｜｜〈1 for｜z｜〈max{1/2＋｜P｜,（1-｜p｜）/2^1/2andφ_P ∈Aut（B^n） such thatφ_（p）=0. As corollaries of the above estimate, we obtain some sharp Bohr＇s type modulus inequalities. In particular, when n=1 and ｜P｜→1, then our theorem reduces to a classical result of Bohr.     

某个用Carlson-Shaffer算子定义的解析函数子类的性质(英文)

Let Hn（p）be the class of functions of the form f（z）=z p＋ ＋∞ Σ k=n akzk＋p,which are analytic in the open unit disk U={z：｜z｜1}.In the paper,we introduce a new subclass Bn（μ,a,c,α,p;φ）of Hn（p）and investigate its subordination relations,inclusion relations and distortion theorems.The results obtained include the related results of some authors as their special case.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

DISTORTION THEOREMS FOR BLOCH MAPPINGS ON THE UNIT POLYDISC Dn

In this article,we establish distortion theorems for some various subfamilies of Bloch mappings defined in the unit polydisc D n with critical points,which extend the results of Liu and Minda to higher dimensions.We obtain lower bounds of | det(f′(z))| and det(f′(z)) for Bloch mapping f.As an application,some lower and upper bounds of Bloch constants for the subfamilies of holomorphic mappings are given.     

Weighted a priori estimates for solution of (−Δ)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis> = <Emphasis Type="Italic">f</Emphasis> with homogeneous dirichlet conditions

Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω  Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap.     

Meromorphic Convex Functions with Negative Coefficients

Let σ_k(a) be the class of functions f(f)=1/z-sur from n=1 to ∞(|a_n|z~n), regular in the punctured disk E={z:0<|z|<1} and satisfying Re(1 zf"(z)/f'(z))<-a (0≤a<1) for z∈E. In this paper we obtain coefficient inequalities, distortion and closure Theorems for the class σ_k(a). Further we obtain the class preserving integral operator of the form     

Properties on solutions of some q-difference equations

We consider the properties on solutions of some q-difference equations of the form ∑ n j=0 aj（z）f（qj z）=an＋1（z）, where a0（z）,..., an＋1（z） are meromorphic functions, a0（z）an（z） ≠ 0 and q ∈ C such that 0 〈 ｜q｜ ≤ 1. We give estimates on the upper bound for the length of the gap in the power series of entire solutions of （＊） when the coefficients a0（z）,..., an＋1（z） are polynomials and 0 〈 ｜q｜ 〈 1. For some special cases, we give estimates of growth of f（z）. And we also show that the case 0 〈 ｜q｜ 〈 1 is different from the case ｜q｜=1.     

Normal Functions Concerning Shared Values

In this paper we discuss normal functions concerning shared values. We obtain the follow result. Let F be a family of meromorphic functions in the unit disc A, and a be a nonzero finite complex number. If for any f ∈F, the zeros of f are of multiplicity, f and f′ share a, then there exists a positive number M such that for any f∈F1（1-｜z｜^2） ｜f′（z）｜/1＋｜f（z）｜^2≤M.     

MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

In this note, the author introduces some new subcIasses of starlike mappings S^＊Ωn1p2,…,pn（β,A,B）={f∈H（Ω）：｜itanβ＋（1-itanβ）2/p（z）аp/аz（z）Jf^-1（z）f（z）-1-AB/1-B^2｜〈B-A/1-B^2},on Reinhardt domains Ωn1p2,…,pn=z∈C^n：｜z1｜^2＋n∑j=2｜zj｜^pj〈1}where - 1≤A〈B〈1,q=min{p2,…,pn}≥1,l=max{p2,…,pn}≥2 and β ∈（-π/2,π/2）.Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator F（z）=（f（z1）＋f＇（z1）Pm（z0）,（f＇（z1））^1/mz0）＇where f is a normalized biholomorphic function on the unit disc D, z = （z1,z0） ∈Ωn1p2,…,pn,z0=（z2,…,zn）∈ C^n-1.Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type/3 and order β These results generalize the modified Roper-Suffridge extension oper-ator from the unit ball to Reinhardt domains. Notice that when p2 = p3 …=pn = 2,our results reduce to the recent results of Feng and Yu.     

由卷积和从属关系刻画的单叶调和函数某些子类

Let SH be the class of functions f = h + ˉg that are harmonic univalent and sensepreserving in the open unit disk U = {z ∈ C : |z| 1} for which f(0) = f′(0)-1 = 0. In the present paper, we introduce some new subclasses of SH consisting of univalent and sensepreserving functions defined by convolution and subordination. Sufficient coefficient conditions,distortion bounds, extreme points and convolution properties for functions of these classes are obtained. Also, we discuss the radii of starlikeness and convexity.     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

THE FEKETE-SZEG ¨O PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.     

A Covering Lemma on the Unit Sphere and Application to the Fourier-Laplace Convergence

A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as follows： Theorem Suppose f ∈ L（En-1）, n≥ 3. If f satisfies the condition 1/θ^n-1∫D（x,θ）｜f（y）-f（x）｜dy=O（1/｜logθ｜）,as θ→0＋, at every point x in a set E of positive measure in Σn-1, then the Cesàro means of critical order ,n-2/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.     

Vilenkin-Like系统的不等式

For bounded Vilenkin-Like system, the inequality is also true：
（∑ k=1 ^∞ kp-2｜f^^（k）｜^p）^1/p ≤ C｜｜f｜｜Hp, 0 〈 p ≤ 2,
where f^^（·） denotes the Vilenkin-Like Fourier coefficient of f and the Hardy space Hp（Gm） is defined by means of maximal functions. As a consequence, we prove the strong convergence theorem for bounded Vilenkin-Like Fourier series, i.e.,
（∑ k=1 ^∞ k^p-2｜｜Skf｜｜p^p）^1/p≤C｜｜f｜｜Hp,0〈p〈1.     

Hua’s theorem with <Emphasis Type="Italic">s</Emphasis> almost equal prime variables

It is proved unconditionally that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented as the sum of s almost equal k-th powers of prime numbers for 2 ≤ k ≤ 10 and s =2k ＋ 1, which gives a short interval version of Hun＇s theorem.     

Approximation by semigroup of spherical operators

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the （n ＋ 1）- dimensional Euclidean space for n ≥2. We prove that such operators form a strongly continuous contraction semigroup of class （l0） and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ＋Vt^γ and the rth Boolean of the generalized spherical Weierstrass operator ＋Wt^k for integer r ≥ 1 and reals γ, k∈ （0, 1] have errors ｜｜＋r Vt^γ- f｜｜X ω^rγ（f, t^1/γ）X and ｜｜＋rWt^kf - f｜｜X ω^2rk（f, t^1/（2k））X for all f ∈ X and 0 ≤t ≤2π, where X is the Banach space of all continuous functions or all L^p integrable functions, 1 ≤p ≤＋∞, on S^n with norm ｜｜·｜｜X, and ω^s（f,t）X is the modulus of smoothness of degree s 〉 0 for f ∈X. Moreover, ＋r^Vt^γ and ＋rWt^k have the same saturation class if γ= 2k.     

On the Extension Operator in Banach Spaces

In [1], Roper and Suffridge introduced an extension operator. This operator is defined for normalized locally univalent function f on the unit disc U = {z∈C：｜z｜〈1} in C by Фn（f）（z）=（f（z1）,√ f′（z1）z0）, where z = （z1,z0） belongs to the unit ball Bn in Cn, z1∈U, z0 = （z2,...,zn） ∈ C^n-1.     

Bloch constant of holomorphic mappings on the unit polydisk of C~n

In this paper,we give a definition of Bloch mappings defined in the unit polydisk D~n, which generalizes the concept of Bloch functions defined in the unit disk D.It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables.We shall establish the corresponding distortion theorems for subfamiliesβ(K)andβ_(loc)(K) of Bloch mappings defined in the polydisk D~n,which extend the distortion theorems of Liu and Minda to higher dimensions.As an application,we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloeh mappings defined in D~n.In particular,our results reduce to the classical results of Ahlfors and Landau when n=1.     

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Let μ be the n-dimensional Marcinkiewicz integral and μb the multilinear commutator of μ. In this paper, the following weighted inequalities are proved for ω ∈ A∞ and 0 〈 p 〈 ∞,
｜｜μ（f）｜｜LP（ω）≤C｜Mf｜LP（ω） and ｜｜μb（f）｜｜LP（ω）≤C｜｜ML（log L）^1/r f｜｜LP（ω）.
The weighted weak L（log L）^1/r -type estimate is also established when p=1 and ω∈A1.