THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

THE FEKETE-SZEG? INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES

Let C be the familiar class of normalized close-to-convex functions in the unit disk.In [17],Koepf demonstrated that,as to a function ■ in the class C,■By applying this inequality,it can be proven that ‖a₃|-|a₂‖≤ 1 for close-to-convex functions.Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.     

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN Cn

In this article,first,a sufficient condition for a starlike mapping of order α f(x) defined on the unit ball in a complex Banach space is given. Second,the sharp estimate of the third homogeneous expan...     

GROWTH AND DISTORTION THEOREMS FOR ALMOST STARLIKE MAPPINGS OF COMPLEX ORDER λ

In this article, the sharp growth theorem for almost starlike mappings of complex order λ is given firstly. Secondly, distortion theorem along a unit direction is also established as the application of the growth theorem. In particular, using our results can reduce to some well-known results.     

SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article,first,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order α on the unit ball in complex Banach spaces are given.Second,the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C~n are also established.In particular,the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings(include quasi-convex mappings of type A and quasi-convex mappings of type B) in several complex variables are get accordingly.Our results state that a weak version of the Bieberbach conjecture for quasi-convex mappings of type B and order α in several complex variables is proved,and the derived conclusions are the generalization of the classical results in one complex variable.     

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper.From the maximum modulus principle of holomorphic functions,we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball B~n in C~n and on bounded and complete Reinhardt domains.Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β,strong and almost starlikeness of order α,strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.     

DIRICHLET FORMS AND SYMMETRIC DIFFUSIONS ON A BOUNDED DOMAIN IN R~d

Let D be a bounded C~3-domain in R~d and(a_(ij))be a bounded symmetric matrixdefined on D.Consider the symmetric form(u,v)=1/2∫_D a_(ij)(x)(u(x))/(x_i) (v(x))/(x_j)dx,u,v∈H~1(D).Under some assumptions it is shown that the diffusion process associated with the regularDirichlet space(,(H~1(D))on L~2(D)can be characterized as a unique solution of acertain stochastic differential equation.     

ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN H~n（-1）

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first kth eigenvalues independent of the domains.     

BLOW-UP AND GLOBAL EXISTENCE FOR A COUPLED SYSTEM OF DEGENERATE PARABOLIC EQUATIONS IN A BOUNDED DOMAIN

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut=△ul up1vq1 and vt=△vm up2vq2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1-(l-P1)(m-q2), the initial data, and the domainΩ.     

ON THE SPECTRUM OF THE TRANSPORT OPERATOR FOR A BOUNDED CONVEX BODY

The eigenvalue problem of the transport operator for a general bounded convex body V is studied mathematically. Where either V or the eigenfunction of the operator is not necessarily of spherical symmetry.It is shown that the spectrum of the operator consists of pure eigenvalues, possibly plusthe point of negative infinity.There is a countable infinity of real eigenvalues accumulating at minus infinity. Each eigenvalue, especially one in the left half-plane, is of index one, There is no complex (non-real) eigenvalue in the right half-plane Re λ>Σ. The solution of the corresponding initial-value problem is also discussed.     

THE PARAMETRIC REPRESENTATION FOR SPIRAL-LIKE MAPPINGS OF TYPE α ON BOUNDED BALANCED PSEUDOCONVEX DOMAINS

LetΩbe a bounded balanced pseudoconvex domain in Cn that admits a plurisubhamonic defining function of class C2. The authors prove that if f is a spiral-like mapping of typeαonΩ, then f can be expressed as a limiting form concerning a Schwarz mapping. Moreover, the characteristics for star-like mappings and spiral-like mappings of typeαare obtained. As a direct application, the growth theorem for spiral-like mappings of typeαis set up.     

ON A SURJECTIVITY FOR THE SUM OF TWO MAPPINGS OF MONOTONE TYPE

In this paper the sum(T+S)of two nonlinear mappings is considered,where T ismaximal monotone or generalized pseudomonotone and S is generalized pseudomonotone orof type(M).By using the concepts of T-boundedness,T-generalized pseudomonotonemappings and mappings of type T--(M) introduced by the author,it is proved that(T+S) is of type (M).A new surjectivity result for multivalued pseudo A-propermappings is given.As a consequence,it is obtained that the coercive mappings of type(M)whose effective domain contains a dense linear subspace are surjectivity. In particular,the author answers affirmatively a part of Browder's question(see[1],p.70).     

THE BLOCH CONSTANT OF LOCALLY BIHOLOMORPHIC MAPPINGS ON BOUNDED SYMMETRIC DOMAINS

51.IntroductionIn[11,westudiedtheboundsofBlochconstantofholomorphicmappingsonirreducibleboundedsymmetricdomains.ThispaperisasuccessivepaPerof[1].WestudytheboundsoftheBlochconstantoflocallybiholomorphicmappingsonirreducibleboundedsymmetricdomains.AllthenotationsofthispaPerarethesameasthoseweusedin[1].LetDbeadomain,whichcontainstheorigin.AholomorphicmappingonDisdefinedasaBlochmappingifthefamilyofmappingsFj={g:g(z)=f(rk(z))-j(gh(o)),forallrkEAut(D)},isanormalfamily.TheBlochnormofaBlochmapp…     

A NECESSARY AND SUFFICIENT CONDITION THAT BIHOLOMORPHIC MAPPINGS ARE STARLIKE ON A CLASS OF REINHARDT DOMAINS

This paper studies the Reinhardt domains B defined as(?)The Schwartz lemma for B is established.Using it the authors give a necessary andsu fficient condition that a local biholomorphic mapping from B to C~n is starlike.It isreduced to the Suffridge's theorem in the case p_1=p_2=…=p_n>1.     

ON THE REGULARITY AND EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

Let Ω(?)R~n be a bounded domain with a smooth boundary (?)Ω L a strictly elliptic operator and c(x)≥0 in Ω. In this paper we are concerned with the following Dirichlet problem with the growth condition (P_1): a<2, for n=2. It is proved that if p(x, t) has all derivatives up to order l which are locally Hlder continuous in (?)×R. and if a_(ij)(x) ∈C_(l 1,α)(Ω) and c(x)∈C_(l,α)(Ω), then any weak solution in W_0~(1,2) of (1) lies in C_(l 2,α)(Ω). Moreover, under the growth condition (P_1) and some additional assumptions, the existence of nontrivial solution of (1) is proved. The main difficulity here is that the simple bootstrapping procedure fails to apply directly to the case of the growth condition (P_1).     

A NOTE ON THE EXISTENCE AND NONEXISTENCE OF GLOBALLY BOUNDED CLASSICAL SOLUTIONS FOR NONISENTROPIC GAS DYNAMICS

Existence of globally bounded classical solution for nonisentropic gas dynamics system has long been studied, especially in the case of polytropic gas. In [4], Liu claimed that sufficient condition has been established. However, the authors find that the argument he used is not true in general. In this article, the authors give a counter example of his argument. Hence, his claim is not valid. The authors believe that it is difficult to impose general conditions on the initial data to obtain globally bounded classical solution.     

A NOTE ON MEYER-KONIG AND ZELLER OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

In the present note we give the correct and improved estimate on the rate of convergence of integrated Meyer-Konig and Zetter operators for function of bounded variation.     

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     

α-CARLESON MEASURE AND BLOCH FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS OF C~n

α－ＣＡＲＬＥＳＯＮＭＥＡＳＵＲＥＡＮＤＢＬＯＣＨＦＵＮＣＴＩＯＮＳＯＮＢＯＵＮＤＥＤＳＹＭＭＥＴＲＩＣＤＯＭＡＩＮＳＯＦＣ~ｎＯｕｙａｎｇＣａｉｈｅｎｇ（欧阳才衡）（ＷｕｈａｎＩｕｓｔ．ｏｆＭａｔｈ．Ｓｃｉ．，ＣｈｉｎｅｓｅＡｃａｄ．ｏｆＳｃｉ．，...     

ON THE BOUNDEDNESS AND THE STABILITY RESULTS FOR THE SOLUTION OF CERTAIN FIFTH ORDER DIFFERENTIAL EQUATIONS

ＯＮＴＨＥＢＯＵＮＤＥＤＮＥＳＳＡＮＤＴＨＥＳＴＡＢＩＬＩＴＹＲＥＳＵＬＴＳＦＯＲＴＨＥＳＯＬＵＴＩＯＮＯＦＣＥＲＴＡＩＮＦＩＦＴＨＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＣＥＭＩＬＴＵＮＣ（ＹｕｚｕｎｃｕＹｉｌＵｎｉｖｅｒｓｉｔｙ，Ｅｇ...