On Growth of Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n which does not vanish in |z| k, k ≥ 1.It is known that for each 0 ≤ s n and 1 ≤ R ≤ k,M (P~(s), R )≤( 1/(R~s+ k~s))[{d~((s)/dx(s))(1+x~n)}_(x=1)]((R+k)/(1+k))~nM(P,1).In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial P(z).     

Some Integral Mean Estimates for Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ k. Fork = 1,it is known that for each r 0 and |α|≥ 1,n(|α|- 1) {∫2π0|P(eiθ)|rdθ}1/r 0r≤ {∫2π0|1+ eiθ|rdθ}1/rmax|z|=|Dα P(z)|.In this paper, we shall first consider the case when k ≥ 1 and present certain generalizations of this inequality. Also for k ≤ 1, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.     

ON A CONJECTURE OF F. NEVANLINNA CONCERNING DEFlCIENT FUNCTION (Ⅱ)

The paper proves on the basis of [1] the following theorem: Let f(z) be an entire function of lower order μ<∞, and a_l(z)(l=1, 2, …, k) be merommphic functions which satisfy T(r, a_     

DISTORTION ESTIMATES OF SYMMETRIC POINTS OF QUASICIRCLE

Let ξ=f(z) be a K-quasiconformal mapping (K-Q.C.) in the complex plane, f(∞)=∞. The image of the real axis under ξ=f(z) is a quasicirclz and the points ξ=f(z) and ξ=f(z) are a pair of symmetric points of the circle, we know that the connection between the quasicircle and the quasiconformal extention is essential. The distortion estimates of the quasicircle and of the symmetric points are both interesting. It is known that there exist two constants C_1(K) and C_2(K), depending on K oniy, such that the inequality     

SOME PROPERTIES OF CONJUGATE FUNCTIONS

1. Introduction Let u(z)be a real-valued fuaction harmonic in |z|<1. Itwill be said u∈h~p(0相似文献   

超球函数的性质

Let Pn^(a,a)(z)be n-degree hyperspherical polinomials,Qn^(a,a)(z) be the second stort of hyperspherical functions,Ea be an elipse on z-plane.In this paper,we found some inequalitys about Pn^(a,a)(z) and Qn^(a,a)(z)on z-plane,and generalize inequality in paper[2] when a=0,to when a≥0.In addition,we also found two important expressions of the relation.     

Teichmǖller Mappings, Harmonic Maps and Holomorphic Quadratic Differentials

Let M be a smooth,compact(closed and without boundary)surface.Consider the metrics σ(z)|dz|2 and ρ(ω)|dω|2 on M,where z=x+iy and ω=u+iv are conformal coordinates on M.     

Oblique derivative problems for nonlinear elliptic systems with measurable coefficients

1 Problem formulation Let Q be a bounded multiply connected domain in RN and the boundary OQ E C2. W6consider the nonlinear elliptic system of second order equationsUnder certain conditions, system (1) can be reduced to the formwhere u = (ul,' t u.), Da = (u..), DZu = (u:..,), andSuppose that (1) (or (2)) satisfiesCondition C For arbitrary functions u'(x), u'(z) E Cd(~) n W::(Q), Fk(x,u,Du,DZu)(k = 1,' I m) satisfy the conditionswhere 0 < g < 1, u == al - u2, and al:), bit), of*),…     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.     

THE SUPPORT POINTS AND EXTREME POINTS OF THE UNIT BALL OF H'p

Byσthe author denotes the areal measure on the unit disk D=z: |z|<1. Let H'p={f(z): f(z) is analytic in D and 1/π∫D|f(z)|p dσ<+∞}. Let B(H＇p)={f:f∈H＇p and 1/π∫D|f(z)|p dσ1. This article researches the support points and extreme points of B(H'p).     

Hankel and Toeplitz type operators on the unit disk

Let D be the unit disk in the complex plane with the weighted measure dμβ(z) = (β+1)--π(1-|z|2)βdm(z)(β＞-1). Then L2(D,dμβ(z)) = ∞k=0(Aβk Aβk) is the orthogonal direct sum decomposition.In this paper, we define the Hankel and Toeplitz type operators, and study the boundedness, compactness and Sp-criteria for them.     

THE （u＋К）-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let Н be a complex,separable,infinite dimensional Hilbert space,T∈L(Н),(U+κ)(T) denotes the (U+κ)-orbit of T,i.e.,(U+κ)(T)={R^-1 TR:R is invertible and of the form unitary plus compact}.Let Ω be an analytic and simply connected Cauchy domain in C and n∈N,A（Ω,n）denotes the class of operators,each of which satisfies (i) T is essentially normal;(ii)σ(T)=Ω^-,ρF（T）∩σ（T）=Ω;（iii）ind(λ-T)=-n,nul(λ-T)=0,(λ∈Ω）。it is proved that given T1,T2∈A(Ω,n）and c&gt;0,there exists a compact operator K with ||K||&lt;ε such that T1+K∈(u+κ)(T2),this result generalizes a result of P.S.Guinand and L.Marcoux[6,15],Furthermore,the authors give a character of the norm closure of (u+κ)(T),and prove that for each T∈А(Ω,n）,there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.     

ON THE SYMPLECTIC INVARIANTS OF A SUBSPACE OF A VECTOR SPACE~*

Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp_(zv)(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp_(2v)(F) is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F     

ON THE NORMAL FAMILY OF MEROMORPHIC FUNCTIONS

In [1],W.K.Hayman has posed the following interesting problems: (A).Let be a family of holomorphic functions in the domain D.Let p be apositive integer,p≥3; and let a,b be two finite complex numbers,a≠0. If forany function f(z)∈ satisfies the condition f’(z)-a{f(z)}~P≠b,     

ESTIMATE OF d_0/d~* FOR STARLIKE FUNCTIONS

Let S~* be the class of functionsf(z)analytic,univalent in the unit disk|z|<1 andmap|z|<1 onto a region which is starlike with respect to w=0 and is denoted as D_f.Letr_0=r_0(f)be the radius of convexity of f(2).In this note,the author proves the following result:(d_0/d~*)≥0.4101492,where d_0= f(z),d~*=|β|.     

Teichmuller Mappings, Harmonic Maps and Holomorphic Quadratic Differentials

Let M be a smooth, compact (closed and without boundary) surface. Consider the metricsσ(z)|dz|~2 and ρ(ω)|dω|~2 on M, where z=x+iy and ω=u+iv are conformal coordinates onM. For a Lipschitz map ω=ω(z): (M, σ|dz|~2)→(M, ρ|dw|~2), we define the energy density of     

ON THE RATE OF CONVERGENCE OF THE GAMMA OPERATOR FOR FUNCTIONS OF BOUNDED VARIATION

Let f be a function defined on the infinite interval[0,∞).The Gamma operator G_n applied to f isG_n(f(t),x)=∫_0~f(nt)g_n(x/t)(dt)/t,whereg_n(u)=u~ne~(-u)/(n-1)!(see[1],P.207).There are some results of convergence and saturation of G_n     

Some Results on the Polar Derivative of a Polynomial

Let P(z) be a polynomial of degree n and for any complex number α, let Dα P(z) = n P(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.     

TWO-STAGE MULTISPLITTING OF SYMMETRIC POSITIVE SEMIDEFINITE MATRICES

Main resultsTheorem 1 Let A be symmetric positive semidefinite.Let (?) be a diagonally compen-sated reduced matrix of A and Let (?)=σI+(?)(σ>0) be a modiffication(Stieltjes) matrixof (?).Let the splitting (?)=M-(?) be regular and M=F-G be weak regular,where M andF are symmetric positive definite matrices.Then the resulting two-stage method corre-sponding to the diagonally compensated reduced splitting A=M-N and inner splitting M=F-G is convergent for any number μ≥1 of inner iterations.Furthermore,the iteration     

ON SATURATION CLASS FOR JACKSON SUMES OF BESSEL SERIES

Let function f(z) be analytic in |z|<1 and continuous on |z|≤1. The saturation class for the Jackson sums of the Bessel series of f(z) is discussed.