Differential Inequalities and Starlike Functions

We develop sufficient conditions for various types of starlikeness for functions analytic in the unit disk. The problems that we consider are more general than similar problems considered by previous authors. Furthermore we obtain stronger estimates than in earlier works and in several cases sharp ones.     

Lyapunov Functions for Infinite-Dimensional Systems

We study Lyapunov functions for infinite-dimensional dynamical systems governed by general maximal monotone operators. We obtain a characterization of Lyapunov pairs by means of the associated Hamilton-Jacobi partial differential equations, whose solutions are meant in the viscosity sense, as evolved in works of Tataru and Crandall-Lions. Our approach also leads to a new sufficient condition for Lyapunov pairs, generalizing a classical result of Pazy.     

On the Coefficients of Several Classes of Bi-Univalent Functions

In this paper, we investigate the bounds of the coefficients of several classes of bi-univalent functions. The results presented in this paper improve or generalize the recent works of other authors.     

几类亚纯双单叶函数的系数估计

研究了几类亚纯双单叶函数的系数估计,所得结果改进或推广了部分作者的结论.     

关于某类p叶解析函数

引进了新的p叶解析函数子类Bp,λ,α(A,B,C,D),应用微分从属方法证明它的从属关系、包含关系、偏差定理和不等式性质.     

An Unified Approach to Second Hankel Determinant of Bi-Subordinate Functions

In this work, we obtained the second Hankel determinant with a different method for a certain class of analytic bi-univalent function which is defined by subordinations in the open unit disk. Moreover, the presented results in this work improve or generalize the recent works of other authors.     

Algebraic Representation of Correlation Functions in Integrable Spin Chains

Taking the XXZ chain as the main example, we give a review of an algebraic representation of correlation functions in integrable spin chains obtained recently. We rewrite the previous formulas in a form which works equally well for the physically interesting homogeneous chains. We discuss also the case of quantum group invariant operators and generalization to the XYZ chain. Communicated by Vincent Rivasseau Dedicated to the memory of Daniel Arnaudon Submitted: January 18, 2006; Accepted: February 28, 2006     

A Knowledge Base on Special Functions

An approach to developing a mathematical knowledge base on special functions is described and an example of such a knowledge base is presented. The problem consists of creating a uniform structural approach to classification of special functions of different types. Such an approach was realized in a series of papers by the authors. Two variants of an experimental version of the SFTools knowledge base developed on the basis of such a classification are presented. The first variant is realized in the Delphi language and works in the environment of the Windows 95 and Windows NT operating systems. The second variant presents an Internet modification of SFTools; it is realized in the Java language. Bibliography: 6 titles.     

Generalized Invariant Monotonicity and Invexity of Non-differentiable Functions

This paper is devoted to the study of relationships between several kinds of generalized invexity of locally Lipschitz functions and generalized monotonicity of corresponding Clarke’s subdifferentials. In particular, some necessary and sufficient conditions of being a locally Lipschitz function invex, quasiinvex or pseudoinvex are given in terms of momotonicity, quasimonotonicity and pseudomonotonicity of its Clarke’s subdifferential, respectively. As an application of our results, the existence of the solutions of the variational-like inequality problems as well as the mathematical programming problems (MP) is given. Our results extend and unify the well known earlier works of many authors.     

On the Coefficients of Several Classes of Bi-univalent Functions Defined by Convolution

In this paper, we introduce several new subclasses of the function class Σ of bi-univalent functions analytic in the open unit disc defined by convolution.Furthermore, we investigate the bounds of the coefficients |a2| and |a3| for functions in these new subclasses. The results presented in this paper improve or generalize the recent works of other authors.     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

凸函数的几个Hadamard型不等式

对于凸函数建立了几个新的 Hadamard型不等式 ,比如f ∑nk=1qkak∫A+yA- yg(x) dx ∫A+yA- yf (x) g(x) dx ∑nk=1qkf (ak) ∫A+yA- yg(x) dx和f (pa +qb) p∫ξaf (x) g(x) dx∫ξag(x) dx+q∫bξf (x) g(x) dx∫bξg(x) dx pf (a) +qf (b)等 ,推广了前人所做的工作 .     

一类由切比雪夫多项式和$q$-微分算子定义的双单叶解析函数的初始系数估计

本文介绍了一类由切比雪夫多项式和$q$-微分算子定义的双单叶解析函数,得到了相关系数估计和Fekte-Szego不等式,所得结果改进或推广了已有部分作者的结论.     

A Framework for Globally Convergent Algorithms Using Gradient Bounding Functions

This paper presents a novel framework for developing globally convergent algorithms without evaluating the value of a given function. One way to implement such a framework for a twice continuously differentiable function is to apply linear bounding functions (LBFs) to its gradient function. The algorithm thus obtained can get a better point in each iteration without using a line search. Under certain conditions, it can achieve at least superlinear convergent rate 1.618 without calculating explicitly the Hessian. Furthermore, the strategy of switching from the negative gradient direction to the Newton-alike direction is derived in a natural manner and is computationally effective. Numerical examples are used to show how the algorithm works.     

关于一类含有Dziok-Srivastava算子的多叶解析函数的包含关系和卷积性质

该文引进和研究了一类含有Dziok-Srivastava算子的新的多叶解析函数. 得到了这一函数类中的一些有趣的性质, 如包含关系, 卷积性质等. 这些结果改进和拓展了早期的一些工作. 同时也得到了其他一些新的结果.     

The Starlikeness of Analytic Functions of Koebe Type with Complex Order

For α 0, λ 0 and β,η∈R, we consider the M(α,λ)_b of normalized analyticα—λ convex functions defined in the open unit disc U. In this paper we investigate the class M(α, λ,β,η)_b,with f_b := z/(1-z~n)~b being Koebe type. By making use of Jack's Lemma as well as several differential and other inequalities, the authors derive sufficient conditions for starlikeness of the class M(α, λ, β, η)_b of n-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in earlier works are also indicated.     

Entire Approximations for a Class of Truncated and Odd Functions

We solve the problem of finding optimal entire approximations of prescribed exponential type (unrestricted, majorant and minorant) for a class of truncated and odd functions with a shifted exponential subordination, minimizing the $L^{1}(\mathbb {R})$ -error. The class considered here includes new examples such as the truncated logarithm and truncated shifted power functions. This paper is the counterpart of the works (Carneiro and Vaaler in Trans. Am. Math. Soc. 362:5803–5843, 2010) and (Carneiro and Vaaler in Constr. Approx. 31(2):259–288, 2010) where the analogous problem for even functions was treated.     

Sharp Inequalities for BMO Functions

The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 p ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ : I → R and any subset E I of positive measure,For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ~p and exponential bounds for the BMO class.     

Refinements on Gap Functions and Optimality Conditions for Vector Quasi-Equilibrium Problems via Image Space Analysis

By means of some new results on generalized systems, vector quasi-equilibrium problems with a variable ordering relation are investigated from the image perspective. Lagrangian-type optimality conditions and gap functions are obtained under mild generalized convexity assumptions on the given problem. Applications to the analysis of error bounds for the solution set of a vector quasi-equilibrium problem are also provided. These results are refinements of several authors’ works in recent years and also extend some corresponding results in the literature.     

Maximal Unstable Dissipative Interval to Preserve Multi-scroll Attractors via Multi-saturated Functions

In this paper, we present families of piecewise linear systems which are controlled by a continuous piecewise monoparametric control function for the generation of monoparametric families of multi-scroll attractors. Thus, the maximum range of values that the parameter set can take in order to preserve the useful dynamics for generating of multi-scroll attractors is found and it will be called maximal robust dynamics interval. This class of dynamical systems is the result of combining two or more unstable “one-spiral” trajectories. We give necessary and sufficient conditions in order to preserve multi-scroll attractors in terms of a parameter, i.e., a family of multi-scroll attractors is generated by means of a family of switching systems with multiple monoparametric companion matrices. Lastly, we provide an example to show how the developed theory works.