The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

极大高阶交换子的端点估计

该文主要确立了当b∈BMO 时, 极大高阶奇异积分算子交换子T_{b, m}* 满足如下不等式 |{y∈Rⁿ:T_{b, m}*f(y)>λ}|≤C||b||^m_BMO∫_Rⁿ|f(y)|/λ (1+log⁺|f(y)|/λ)^mdy 且T_{b, m}* 在L^p(Rⁿ)(1 < p <∞上有界.     

Weighted Modular Inequalities for Hardy Type Operators

Given weight functions , w, and v, the weighted modular inequality is characterized. Here Qis a strictly increasing function with Q(0) = 0, Q() = and2Q(x) Q(C x), P is a Young's function, and T is the Hardy operatoror a Hardy type operator. In particular, a characterizing conditionfor the Hardy type operator to map L^p(w) to L^q(v) when 0 <q < 1 p < is deduced. In addition, a new proof for theMaz'ja-Sinnamon theorem is given, and weighted Lorentz norminequalities for Hardy type operators are established. 1991Mathematics Subject Classification: primary 26D15, 42B25; secondary26A33, 46E30.     

Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

Let U(λ, μ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

一类高阶线性微分方程解的复振荡

In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

非齐次粗糙核参数型Marcinkiewicz算子的H~p有界性

设Ω是球面上函数,b是径向函数,ρ是实部正的复数;设Ψ为C~2([0,∞))的递增凸函数,Ψ(0)=0.本文研究非齐次粗糙核参数型Marcinkiewicz算子μ_(Ω,b)~ρ,以及旋转曲面上的非齐次粗糙核参数型Marcinkiewicz算子μ_(Ω,Ψ,b)~ρ,给出非齐次粗糙核Ω和b的最小光滑性条件,建立算子μ_(Ω,b)~ρ和μ_(Ω,Ψ,b)~ρ在Hardy空间和弱Hardy空间上的有界性.本文结果推进了先前b≡1情形的已有工作.     

一类两点边值问题的多重非负解

该文考虑两点边值问题[1/q(t)][q(t)y′(t)]′+p(t)f(y(t))= 0,λ_1 y(α)-λ_2y′(α)=0 and y(β)=B非负解的存在性, 其中p(t)可能在t=α或t=β附近具有奇异性, f(0)≥0, lim_(y→+∞)f(y)/y=+∞, 并且存在y>0, 使得f(y)<0.      

A property of entire functions with real taylor coefficients

Suppose thatf(z) is an entire transcendental function with real Taylor coefficients, M(r)= max¦f(z)¦on¦z¦=r, and {_n} is the sequence of sign changes of the coefficients. We will show that if (1/_n)<, then .Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 395–402, September, 1975.The author thanks A. A. Gol'dberg for his useful comments.     

Flow of a nematic liquid crystal near the leading edge of an infinite prism

Flow of a nematic liquid crystal in an infinite wedge boundedby sidewalls = ± (with no-slip condition) is considered.The fluid is contained in the region 0 r < , – and – < z < (0 ). The near-tip velocity fieldis assumed to have the form v_i(r, ) = rF_i()(i = r, , z) as rtends to zero. We investigate the dependence of eigenvalues and functions F_i() on the tilt angle, G(), between the directorfield and the plane z = c (c ) and on the included angle 2 of the wedge shaped prism. Two kinds of nematicliquid crystal are considered as examples: MBBA and PAA near25 °C and 125 °C, respectively. In general, when 0 <G() < /2 the liquid crystalline material is curvilinear anisotropicand no symmetry properties are found. Here all velocity fieldcomponents are coupled. This coupling reduces the magnitudeof the leading-order eigenvalue and the one with smallest realpart is purely real for any wedge included angle. However, complexeigenvalues can occur for the next eigenvalues ordered in termsof the magnitude of the real part. Thus, if we impose the appropriatebehaviour on the far velocity field so that it is orthogonalto the eigenvectors associated with the first real eigenvalues,the remaining flow fields may display eddies.