Landau定理中的一个新上界

设f(z)=a₀+a₁z+…是单位圆内不取0与1的全纯函数.著名的Landau定理断言:存在一个与f无关的常数C使得|a₁|≤2|a₀|{|log|a₀||+C}.曾有许多数学家致力于估计此常数C.最后在1980年前后,由赖万才、Hempel及Jenkins独立地给出C的精确值:C=Γ⁴(1/4)/4π².本文的主要目的是要建立一种新形式的上界估计式以表明上述结果仍可改进.本文证明了: |a₁|≤2|a₀|{|log|a₀||+C-M|a₀ⁿ+1|²},其中M是一个与f无关的正的常数,而n取值为-1或1,依照|a₀|>1或≤1而定.     

Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification*

Let M be an open Riemann surface and G : M → Pⁿ(C) be a holomorphic map. Consider the conformal metric on M which is given by ds² = k e Gk^2m|ω|², where eG is a reduced representation of G, ω is a holomorphic 1-form on M and m is a positive integer. Assume that ds² is complete and G is k-nondegenerate(0 ≤ k ≤ n). If there are q hyperplanes H₁, H₂, · · · , H_q Pⁿ(C) located in general position such that G is ramified over H_j with multiplicity at least γ_j (> k) for each j ∈ {1, 2, · · · , q}, and it holds that Xj=1 1 -k/γ_j> (2n - k + 1) （mk/2+ 1）,then M is flat, or equivalently, G is a constant map. Moreover, one further give a curvature estimate on M without assuming the completeness of the surface.     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

关于富里埃及数和幂级数的蔡查罗平均

Let f(x) be a periodic function of period 2π,|f(x)|and |f(x)|^p integrable on [0,2π].It is said to be f∈H_X¹,if (∫₀^2π|f(x+h)-f(x)|^pdx)^1/p≤|h|.It is said to be f∈H_X²,if (∫₀^2π|f(x+h)+f(x-h)-2f(x)|^pdx)^1/p≤2|h|.     

p-值星形函数的子类

Let M_n+p-1 denote the class of functions f(z) = 1/z^p+a₀/z^p-2+a₁/z^p-2+…+a_n+p-1zⁿ+…, regular and p-valent in the annulus 0<|z|<1 and satisfying Re((D^n+p f(z))/(D^n+p-1 f(z)))-2)<-(n+p-1)/(n+p),|z|<1,n>-p where D^n+p-1 f(z)=1/z^p((z^n+2p-1f(z))/(n+p-1)!)^(n+p-1).M_n+p?M_n+p-1 is proved. Since M₀ is the subclass of p-valent meromorphically starlike functions, all functions in Mn + p-1 are p-valent meromorphically star-like functions. Further the integrals of functions in M_n+p-1, are considered.     

关于加权位移的酉等价问题

In this paper, we study the unilateral weighted shift which is unitarily equivalent to a Toeplitz operator and prove a similar result to that in [ 1 ] without the hypothesis that the shift must be hyponormal. As a corollary, we show that if the weight sequence {a_n}_n=0^∞ of the shift is convergent, then 1-a_n²=(1-a₀²)ⁿ⁺¹ ?n≥0     

NA随机场对数律的收敛速度

设d是一个正整数, N ^d是d -维正整数格点.设{X_n , n∈N ^d} 是一同分布的负相伴随机场, 记S_n =∑_{k≤ n} X_k, S_n^(k)=S_n-X_k, 如果r >2, EX₁ = 0 和σ²= Var(X₁}, 则存在一个正数M:=100√(r-2)(1+σ²)使得下列条件等价 (I)E |X₁|^r (log|X₁|)^d-1-r/2 <∞; (II)∑_{n∈ N^d} |n|^r/2-2P(max_{1≤ k≤ n} |S_n^(k)|≥ (2^d+1 )ε√|n| log |n |) <∞,∨ε > M; (III)∑_{n∈N ^d} |n|^r/2-2P(max_{1≤ k≤n} |S_k |≥ε√| n} log| n |) <∞,∨ε > M. (III)\ \ $\sum\limits_{{{\bf n}}\in {{\cal N}}^{d}} |n|^{r/2-2} P(\max\limits_{{\bf 1}\leq{\bf k}\leq{\bf n}}|S_{{\bf k}}|\geq \varepsilon \sqrt{|{\bf n}|\log |{\bf n}|})<\infty$, $\forall\varepsilon>M$.     

Semiamart与Gaft类的收敛性

Let (Ω,F,P) be a probability space,(F_n) be an increasing sequence of sub-σ-fields of F, a sequence (x_n) of integrable random variables adapted to (F_n), if (x_n) is progressive uniformly integrable or(x_n)is a L¹-bounded Semiamart, then ( 1 ) x_m-x_n→0 (Pr.), ( 2 ) E (x_m|F_n)-x_n→0 (pr.) are equivalent is proved in this paper .     

切于已知球的单形宽度

Let w(△_n) denote the width of a non-degenerate simplex △_n in Eⁿ and r(△_n) denote the inradius of the simplex.Then, in this paper, we prove the ine-qunlity as below: Theorem:w(△_n)≤β_nr(△_n) where β_n =(n^1/2(n+1))/([(n+1)/2]^1/2(n+1-[(n+1)/2])^1/2) The equality holds if and only if the simplex is regular.     

线性模型中误差方差估计的指数收敛速度

Suppose given a linear model y_j=x'_jβ+μ_j, j=1,2,…, The random errors all have a mean zero and unknown variance σ², 0<σ²<∞. Let σ_n² be the estimate of σ² based on the residual sum of squares and calculated from (x_j, y_j), j=1,…,n. In this paper we show that if μ₁,μ₂,…, are independent but not necessarily identically distributed, and some further conditions on {μ_j} and (x₁|…|x_n) are satisfied, then for any ε>0 there exist constant ρ_ε, 0<ρ_ε<1, Such that P(|σ_n²-σ²|≥ε)=O(ρ_εⁿ).