RADII OF P-VALENCE OF CERTAIN ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS

In this paper we determine the radii of p-valence of the function F(z)defined byF(z)=(1-λ)f(z) λ/pzf'(z).zεDwhere D={z:|z|<1},λ≥0 and the function f(z)belongs to certain subclassesof analytic p-valent functions with negative coefficients.     

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

The Hilbert boundary value problemRe{λ(t) P√ψ (t)} = c(t), t ∈ Lof normal type with Holder continuous coefficients is discussed, where Lis the unit circle |t| = 1,p ≥ 2 is any definite integer, Ψ (t) is the boundary value of the unknown function Ψ (z) holomorphic in |z| ＜ 1 with single-valued continuous P√ψ (t)} on L.     

EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS AND HARDY TERMS

This paper deals with the existence of solutions to the elliptic equation -△uμu/|x|2=λu |u|2*-2u f(x, u) in Ω, u = 0 on ( a)Ω, where Ω is a bounded domain in RN(N≥3),0∈Ω,2*=2N/N-2,λ＞0,λ(a)σμ, σμ is the spectrum of the operator -△- μI/|x|2with zero Dirichlet boundary condition, 0 ＜μ＜-μ,-μ=(N-2)2/4,f(x,u) is an asymmetric lower order perturbation of |u|2*-1 at infinity. Using the dual variational methods, the existence of nontrivial solutions is proved.     

Some Generalization for an Operator Which Preserving Inequalities Between Polynomials

For a polynomial p(z) of degree n which has no zeros in |z| 1, Dewan et al.,(K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363(2010), 38–41) established zp′(z) +nβ2p(z) ≤n2{( β2 + 1+β2)max|z|=1|p(z)|-( 1+β2- β2)min|z|=1|p(z)|},for any complex number β with |β|≤ 1 and |z| = 1. In this paper we consider the operator B, which carries a polynomial p(z) into B[ p(z)] := λ0p(z) + λ1(nz2)p′(z)1!+ λ2(nz2)2 p′′(z)2!,where λ0, λ1, and λ2are such that all the zeros of u(z) = λ0+c(n,1)λ1z+c(n,2)λ2z2lie in the half plane |z| ≤ |z-n/2|. By using the operator B, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.     

On Some Inequalities Concerning Rate of Growth of Polynomials

In this paper we consider a class of polynomials P(z) = a0+∑n v=t a v z v, t ≥ 1not vanishing in |z|k, k≥1 and investigate the dependence of max|z|=1|P(Rz)-P(rz)on max|z|=1|P(z)|, where 1 ≤ r R. Our result generalizes and refines some know polynomial inequalities.     

ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

Let pj ∈ N and pj≥ 1, j = 2, ···, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =(z1, z′2, ···, z′k)′∈ C × Cn2×···× Cnk: |z1|2+ ||z2||p22+ ··· + ||zk ||pk k 1} given11 by F P′j(zj),(f(z1))p2 z′2, ···,(f′(z1))pk z′k)′, where f is a normaljized biholomorphic function k(z) =(f(z1) + f′(z1)=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj : Cnj-→ C is a homogeneous polynomial of degree pj and zj =(zj1, ···, zjnj)′∈ Cnj, nj ≥ 1, pj ≥ 1,nj1||zj ||j =()pj. In this paper, some conditions for Pjare found under which the loperator p |zjl|pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.     

关于Ahuja的一个结果

In [1], O. P. Ahuja established the following result: If F is an element of R_n(α)for n≥0 and 0≤α<1, F(z)=(c+1)/(z~c)integral from 0 to z f(t)t~(c-1)dt with |z|<1,Rec≥-α, and 0≤β<1. then the function f is an element of R_n(β) for |z|相似文献   

Sharp distortion theorems for some subclasses of star like mappings on Bnp in Cn

We mainly establish the distortion theorems of Jacobi determinant for three subclasses of starlike mappings on Bpn,where Bp={z=(z1,……,zn)T∈Cn:∑nl=1|zl|p<1}p>1.In particular,the above distortion theorems are sharp if Bpn is the unit polydisk in Cn.Our results reduce to the corresponding classical results in one dimension of complex function theory.     

On the existence of diffusions with singular drift coefficient

Let L:be an operator in R~d,where the matrix(a) is bounded,Holder continuous and uniformly positive definite,and (b~t(x)) is Borel measurable.In this paper we prove the existence of L-diffusion under the hypothesis that where g_1(z)=1,g_2(z)=-ln|z|and g_c(z)=|z|~(2-d) for d≥3.     

SOME SUBGROUPS OF THE EXTENDED HECKE GROUPS (λ_q)

The authors consider the extended Hecke groups H(λq) generated by T(z) = -1 / z, S(z) = -1/(z + λq,) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(λq), the second commutator subgroup H (λq) and the principal congruence subgroups Hp(λq) of the extended Hecke groups H(λq) are studied. Also, relations between them are given.     

某类$q$差分方程亚纯解的性质

对于一个有穷非零复数$q$, 若下列$q$差分方程存在一个非常数亚纯解$f$, $$f(qz)f(\frac{z}{q})=R(z,f(z))=\frac{P(z,f(z))}{Q(z,f(z))}=\frac{\sum_{j=0}^{\tilde{p}}a_j(z)f^{j}(z)}{\sum_{k=0}^{\tilde{q}}b_k(z)f^{k}(z)},\eqno(\dag)$$ 其中 $\tilde{p}$和$\tilde{q}$是非负整数, $a_j$ ($0\leq j\leq \tilde{p}$)和$b_k$ ($0\leq k\leq \tilde{q}$)是关于$z$的多项式满足$a_{\tilde{p}}\not\equiv 0$和$b_{\tilde{q}}\not\equiv 0$使得$P(z,f(z))$和$Q(z,f(z))$是关于$f(z)$互素的多项式, 且$m=\tilde{p}-\tilde{q}\geq 3$. 则在$|q|=1$时得到方程$(\dag)$不存在亚纯解, 在$m\geq 3$和$|q|\neq 1$时得到方程$(\dag)$解$f$的下级的下界估计.     

系数的级相同的高阶微分方程解的增长性

In this paper,we consider the growth of solutions of some homogeneous and nonhomogeneous higher order differential equations.It is proved that under some conditions for entire functions F,A_(ji) and polynomials P_j(z),Q_j(z)(j=0,1,…,k-1;i=1,2)with degree n≥1,the equation f~(k)+(A_(k-1,1)(z)e~(p_(k-1)(z))+A_(k-1,2)(z)e~(Q_(k-1(z)))/~f~(k-1)+…+(A_(0,1)(z)e~(P_o(z))+A_(0,2)(z)e~(Q_0(z)))f=F,where k≥2,satisfies the properties:When F ≡0,all the non-zero solutions are of infinite order;when F=0,there exists at most one exceptional solution fo with finite order,and all other solutions satisfy λ(f)=λ(f)=σ(f)=∞.     

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.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

Let A(z) be an entire function with μ(A) 1/2 such that the equation f~((k))+A(z)f = 0, where k ≥ 2, has a solution f with λ(f) μ(A), and suppose that A_1 = A+h,where h■0 is an entire function with ρ(h) μ(A). Then g~((k))+ A_1(z)g = 0 does not have a solution g with λ(g) ∞.     

关于三圈图的拉普拉斯谱半径的一些结果

边数等于点数加二的连通图称为三圈图.~设 ~$\Delta(G)$~和~$\mu(G)$~
分别表示图~$G$~的最大度和其拉普拉斯谱半径,设${\mathcal
T}(n)$~表示所有~$n$~阶三圈图的集合,证明了对于~${\mathcal
T}(n)$~的两个图~$H_{1}$~和~$H_{2}$~,~若~$\Delta(H_{1})>
\Delta(H_{2})$ ~且 ~$\Delta(H_{1})\geq \frac{n+7}{2}$,~则~$\mu
(H_{1})> \mu (H_{2}).$ 作为该结论的应用,~确定了~${\mathcal
T}(n)(n\geq9)$~中图的第七大至第十九大的拉普拉斯谱半径及其相应的极图.     

与正弦函数有关的星形函数逆的Toeplitz行列式

Let S_s*be the class of normalized functions f defined in the open unit■such that the quantity zf’(z)/f(z)lies in an eight-shaped region in the right-half plane and satisfies the condition■.In this paper,we aim to investigate Toeplitz determinants for the inverse of this function classes S_s*associated with sine function.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

Variation of Potentials in the Unit Ball of $ {\shadC}^n$

In this paper, we study the variation of invariant Green potentials G w in the unit ball B of $ {\shadC}^n$ , which for suitable measures w are defined by $$ G_{\mu}(z) = \int_{B}G(z,w)\, d\mu(w), $$ where G is the invariant Green function for the Laplace-Beltrami operator ¨ j on B . The main result of the paper is as follows. Let w be a non-negative regular Borel measure on B satisfying $$ \int_{B}(1-|w|^2)^n\log {1 \over (1-|w|^2)}\, \d\mu(w) ] B , { z denotes the holomorphic automorphism of B satisfying { z (0) = z , { z ( z ) = 0 and ( { z { z )( w ) = w for every w ] B .     

UNIFORM STRONG CONVERGENCE RATE OF NEAREST NEIGHBOR DENSITY ESTIMATION

Based on [3] and [4],the authors study strong convergence rate of the k_n-NNdensity estimate f_n(x)of the population density f(x),proposed in [1].f(x)>0 and fsatisfies λ-condition at x(0<λ≤2),then for properly chosen k_nlim sup(n/(logn)~(λ/(1 2λ))丨_n(x)-f(x)丨C a.s.If f satisfies λ-condition,then for propeoly chosen k_nlim sup(n/(logn)~(λ/(1 3λ)丨_n(x)-f(x)丨C a.s.,where C is a constant.An order to which the convergence rate of 丨_n(x)-f(x)丨andsup 丨_n(x)-f(x)丨 cannot reach is also proposed.