Hardy-Sobolev类上的Bohr型不等式

Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type
‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,
where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.     

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.     

Qp空间中的Jackson定理

在Qp空间上建立了Jackson型不等式,即对任意f（z）=∑j=0 ^∞ αjz^j∈Qp,0≤p〈∞,α〉1及k-1∈N,有 ｜｜f（z）-Г（k）/Г（k＋α）∑ j=0 ^k-1Г（k-j＋α）/Г（k-j）αjz^j｜｜Qp≤C（α）ω（1/k,f,Qp）,其中ω（1／k,f,Qp）为Qp空间中的连续模,C（α）是仅与参数α有关的正常数．     

The Equivalence of BMOA and VMOA on the Unit Ball of C^n

In this paper, we characterize BMOA and VMOA on the unit ball in terms of normsNp(f)=sup/x∈B{1/σ（Q（z))∫Q(z)│f(ξ)-f(z)│pdσ(ξ)}^1/p,1≤p&;lt;∞ and more general Garcia norms.Where Q(z)=Q(n(z),δz)={ξ∈S,│^1/2&;lt;δZ}.n(z)={e1,z=0,/z/│z│,z=0,δz=[2(1-│z│)]^1/2.     

Normal Families and Uniqueness of Entire Functions and Their Derivatives

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.     

SOME CONVOLUTION AND INCLUSION PROPERTIES FOR SUBCLASSES OF MEROMORPHIC p-VALENT FUNCTIONS INVOLVING INTEGRAL OPERATOR

Making use of the linear operator Lp^m（λ,e）f（z）=1/z^p＋∞∑k=1[e/e＋λk]^m akz^k-p,where e〉0,λ〉0,p∈N,m∈N0=NU{0},z∈U^＊ and f（z）∈∑p,we introduce two subclasses of meromorphic p-valent analytic functions and investigate convolution and inclusion properties for these classes.     

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.     

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.     

BOUND STATES FOR A STATIONARY NONLINEAR SCHRODINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R^3△

We study the following Schrodinger-Poisson system where （Pλ）{-△u＋ V（x）u＋λФ（x）u^p=x∈R^3,-△Ф=u^2,lim│x│→∞Ф（x） =0,u〉0,where λ≥0 is a parameter,1 〈 p 〈 ＋∞, V（x） and Q（x）=1 ,D.Ruiz[19] proved that（Pλ）with p∈ （2, 5） has always a positive radial solution, but （Pλ） with p E （1, 2] has solution only if λ 〉 0 small enough and no any nontrivial solution if λ≥1/4.By using sub-supersolution method,we prove that there exists λ0〉0 such that（Pλ）with p ∈（1＋∞）has alaways a bound state（H^1（R^3）solution for λ∈[0,λ0）and certain functions V（x）and Q（x）in L^∞（R^3）.Moreover,for every λ∈[0,λ0）,the solutions uλ of （Pλ）converges,along a subsequence,to a solution of （P0）in H^1 as λ→0     

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.     

带有无界势和一般时间频率的非周期离散非线性Schr#246;dinger方程：无穷多个孤立子

本文研究下面的非周期离散非线性Schrödinger 方程：
-Δu_n + v_nu_n - ωun = g_n（u_n）,n ∈ Z,
其中V = {v_n}_n∈Z 和g_n 都是非周期的,当|n| → +∞ 时,v_n → +∞,并且时间频率ω ∈ R 可以满足下面的任何一种情形：（1）ω 属于算子-Δ + V 的一个有限谱间隔;（2）ω < inf σ（-Δ + V）;（3）ω ∈ σ（-Δ+ V）,其中σ（-Δ+ V）表示-Δ+ V的谱. 本文将用一些局部条件（在无穷远或零处）来代替一些全局条件. 利用变化的喷泉定理,当非线性项在无穷远处是超线性时,本文得到这个方程的无穷多个非平凡孤立子,并且,也得到指数衰减的孤立子的存在性.     

一类单叶解析函数族

设T（（α,β）表示单位圆盘δ内单叶解析且具有下列形式的函数族 ｜（zf′）′/（zf′）′＋（1-2α）｜〈β ,这里f（z）=z＋ z ＋∞∑n=2αnz^n ,县g（0）=g′（0）-1=0,本文研究得到了它的系数估计和偏差结论．     

几类B值拟鞅空间上的原子分解

设1〈P≤2,0〈n≤1,X是P一致可光滑空间的Banach空间,则对每个X值拟鞅f=（fn）n≥0∈pHn^σ（X）存在分解fn=∑k∈Zμkαn^k（n≥0）,并且｜｜f｜｜pHα^σ（X）＋｜｜R（f）｜｜α～inf（∑k∈μk^a）^1/a,这里a^k=（an^k）n≥（k∈Z）是一列（1,α,∞;p）拟鞅原子,并且在L^1中收敛,sup k∈z｜｜a^k＊｜｜n〈∞,（μk）k∈Z∈la是非负实数列．对于拟鞅空间pHa^s（X）和qKn（x）成立类似的结果．此外,利用拟鞅原子分解定理,证明了几个拟鞅不等式．     

关于Hardy-Hilbert型积分不等式

设1/p=1/q≈1：1且P〉1．通过引入一个适当的积分核函数和参数λ（λ〉-1）,创建了一种新型Hardy～Hilbert型积分不等式．证明了其常数因子（p^λ=1＋q^λ＋1）Г（λ＋1）是最佳的,其中Г（x）Г-函数．特别,当p=2时,得到了一种新的Hilbert型积分不等式．作为应用,给出了它的一种等价形式．     

共形映像的Hausdorff 测度及其算法

对于Rⁿ 中满足0 < H^s(K) < ∞ 的任意紧致集K, 我们考虑其在共形映射f 作用下的像集的Hausdorff 测度H^s(f(K)). 本文给出了下面结果:
H^s(f(K)) = H^s(K) · ∫_K |D_xf|^sdμ(x),
其中概率测度μ = (H^s|K/H^s(K)) . 给定满足开集条件的自相似集K, 测度μ 恰好是自相似测度, 因此可以应用上述公式计算f(K) 的Hausdorff 测度, 例如, K 是λ-Sierpinski 地毯, f(z) = z+εz², 其中0 < λ ≤1/4,复数ε 满足|ε| ≤ 0.1. 而此刻f(K) 恰好是自共形集, 因此我们的算法能计算一类特殊的具有非线性结构的自共形集的Hausdorff 测度.     

复合解析函数族分担亚纯函数的正规性

本文推广了Bergweiler的一个正规定则:设α(z)和F分别是区域D上的非常数解析函数与解析函数族,R(z)是一个次数不低于2的有理函数.如果对族F中函数f(z)和g(z),Rof(z)和Rog(z)分担α(z)IM,并且下述条件之一成立:(1)对任意z0∈D,R(z)-α(z0)有至少两个不同的零点或极点;(2)存在z0∈D使得R(z)-α(z0):=P(z)Q(z)仅有一个零点(或极点)β0,同时k=lp(或k=lq),其中l和k分别是f(z)-β0和α(z)-α(z0)在z0处的零点重数,P(z)和Q(z)分别是次数为p和q的互质的多项式,并且α(z0)∈C∪{∞}.那么F在D内正规.     

本征值问题的统一推导方法及讨论

针对一般情形的本征方程X″(x)-2bX′(x)+λX(x)=0结合第一、二、三类齐次边界条件的统一形式,给出有关本征值问题的统一结果,从而可直接利用分离变量法求解2U/t2=a22U/x2+a1u/x+a2t/u+a3u型等含有ux项的泛定方程的定解问题.     

粗糙核带参数的抛物型Marcinkiewicz函数的一个注记

主要研究了带参数的抛物型Marcinkiewicz函数μσΩ,h（f）的L2（（R）n）有界性,用核的分解技术和Fourier变换估计的方法分别在当1＜-γ＜∞,h∈Hγ＇（IR）＋）,Ω∈L（logL）1/γ（Sn-1）条件下和当1〈γ≤∞,h∈△γ（（IR）＋）,Ω∈Llog＋L（Sn-1）条件下,建立了μσΩ,h（f）的L2（（R）n）有界性,并推广了以前学者的结论.     

Rotenberg模型中迁移半群的本质谱

在Lp（1（≤）p＜＋∞）空间中,本文运用半群理论研究了Rotenberg模型中具光滑边界条件的迁移半群的本质谱.采用半群方法和比较算子等方法,证明了对任意的t＞0,s＞0,算子[UH（t）-U0（t）]U0（s）[UH（t）-U0（t）]在Lp（1＜p＜＋∞）在空间中紧和在L1空间弱紧,得到迁移半群VH（t）与V0（t）有相同的本质谱型.     

单位上三角矩阵群的注记(Ⅲ)

设k_(ij)(1≤ij≤n)是给定的正整数,分别记G={ (1 k12a12…k1na1n 0 1…k2na2n…… 0 0…1 )|aij∈Z},R={ (0 k12a12…k1na1n……0 0…k2na2n 0 0…1 )|aij∈Z},本文证明:当G成群且G的上、下中心群列重合时,其相伴Lie环L(G)与Lie环R同构,其中R的Lie积定义为[A,B]=AB-BA.即得到了此时L(G)的矩阵表示.