ON CRITERIA FOR STARLIKENESS AND CONVEXITY OF ANALYTIC FUNCTIONS

In this paper the authors study (1 - γ zf“(z)/f‘(z))/(zf‘(z)/f(z)) as a criteria for starlikeness and convexity. Sharp upper bound of |α2| and of the Fekete-Szegoe functional |α3-μα2^2| is given for a class of analytic functions defined by using this expression.     

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.     

Multi-dimensional versions of a formula of Popoviciu

In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, a formulation is presented which counts the number of non-negative integer solutions of s×(s 1) linear Diophantine equations and it can be considered as a multi-dimensional versions of the formula counting the number of non-negative integer solutions of ax by = n which is given by Popoviciu in 1953.     

Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke＇s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction F, we extend the smoothness of sphere and differentiability of norm function in Banach space.     

A bivariate extension of Bleimann-Butzer-Hahn operator

Let C(R2+ ) be a class of continuous functions f on R2+. A bivariate extension L.(f ,x,y) of BleimannButzer-Hahn operator is defined and its standard convergence properties are given. Moreover, a local analogue of Voronovskaja theorem is also given for a subclass of C(R2+).     

CALCULUS ON CANTOR TRIADIC SET （Ⅱ）—EXCEPTIONAL SET

In this paper the derivative and its exceptional set according to the alternatively jumping function and self-similar function respectively are discussed. And in the appendix, a proof of the statement 1.2(2) of “Calculus on Cantor triadic set (Ⅱ)—derivative” which is important to the discussion of exceptional set is given.     

A BIVARIATE EXTENSION OF BLEIMANN-BUTZER-HAHN OPERATOR

Let C(R_+~2) be a class of continuous functions f on R+2. A bivariate extension L(?)(f,x,y) of Bleimann-Butzer-Hahn operator is defined and its standard convergence properties are given. Moreover, a local analogue of Voronovskaja theorem is also given for a subclass of C(R+2 ).     

由拟从属关系定义的P叶解析函数子类的Fekete-Szeg?问题（英文）

In this paper, a new certain class of р-valent analytic functions with quasisubordination is defined and the Fekete-Szeg? problems for functions belonging to the class are derived. The results presented here provide extensions of those given in some earlier works.     

A REMARK ON KOLMOGOROVS COMPARISON THEOREM

In approximation theory the theorem of Kolmogorov concerning the comparison of derivatives of differentiable functions defined on the real line is well—known. It plays an important role in establishing sharp inequalities between the norms of derivatives of a function. In this note we establish a comparison theorem of Kolmogorov type on a class of functions which are defined on the real line and can be contlnuated analytically in a stripped region containing the real line. As a consequence we have derived an inequality of Landau-Kolmogorov type on this function class, and moreover, we have applied it to get the exact estimation for the Kolmogorov''s N-widths of the analytic function class.     

Efficient estimation of functional-coefficient regression models with different smoothing variables

In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

P-Bloch函数的偏差定理和单叶半径

In this paper, we establish distortion theorems for both normalized p-Bloch functions with branch points and normalized locally univalent p-Bloch functions defined on the unit disk, respectively. These distortion theorems give lower bounds on |f′(z)| and ■f′(z). As applications of these distortion theorems, the lower bounds of the radius of the largest schlicht disk on these Bloch functions are given, respectively. Notice that when p = 1, our results reduce to that of Liu and Minda.     

The Refined Schwarz-Pick Estimates for Positive Real Part Holomorphic Functions in Several Complex Variables

In this article,the refined Schwarz-Pick estimates for positive real part holomorphic functions■are given,where k is a positive integer,and G is a balanced domain in complex Banach spaces.In particular,the results of first order Fréchet derivative for the above functions and higher order Frechet derivatives for positive real part holomorphic functions■are sharp for G=B,where B is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces.Their results reduce to the classic...     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.     

A Boundary Element Approximation of a Signorini Problem with Friction Obeying Coulomb Law

In this work, a Signorini problem with Coulomb friction in two dimensional elasticity is considered. Based on a new representation of the derivative of the double-layer potential, the original problem is reduced to a system of variational inequalities on the boundary of the given domain. The existence and uniqueess of this system are established for a small frictional coefficient. The boundary element approximation of this system is presented and an error estimate is given.     

针对一维空间分数阶扩散方程的一个新的最大模近似公式

Based on the maximum principle,the difference formula defined on a non-integral node is given to approximate the fractional Riemann-Liouville derivative and the finite difference scheme for solving one-dimensional space fractional diffusion equations(FDEs) with variable coefficients is presented.Furthermore,using the maximum principle the scheme is proved unconditionally stable and secondorder accuracy in spatial grid size.Several numerical examples are given to verify the efficiency of the scheme.     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

A BIVARIATE EXTENSION OF BLEIMANN－BUTZER－HAHN OPERATOR

Let C( R ^2) be a class of continuous functions f on R ^2.A bivariate extension Ln(f,x,y)of Bleiman-Butzer-Hahn operator is defined and its standard convergence properties are given.Moreover,a local analogue of Voronovskaja theorem is also give for a subclass of C(R ^2).     

SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

We study the asymptotic behavior near the boundary of u(x,y) = Ky μ(x),defined on the half-space R+ ×RN by the convolution of an approximate identity Ky(.)(y > 0) and a measure μ on RN.The Poisson and the heat kernel are unified as special cases in our setting.We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ.Then a boundary limit theorem of Fatou’s type for singular measures is proved.Meanwhile,the asymptotic behavior of a quotient of Kμ and Kν is also studied,then the corresponding Fatou-Doob’s boundary relative limit is obtained.In particular,some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours.At the end,an application in fractal geometry is given.     

CAUCHY PROBLEM FOR FRACTIONAL BEAM-LIKE EQUATIONS

In one space-and in one time -dimension a beam-like equation is solved, where the second time derivative is replaced by the a?fractional time derivative, 1 <α< 2. The solution is given in closed form in terms of the Mttag-Leffler functions in two parameters.