Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.     

强预拟不变凸函数与强拟不变凸函数

本文提出了两类新的广义凸函数—强预拟不变凸函数与强拟不变凸函数.讨论了强预拟不变凸函数与强拟不变凸函数间的关系,强拟不变凸函数与强伪不变凸函数间的关系.研究了强预拟不变凸函数在多目标优化中的应用.     

Generalized B-vex functions and generalized B-vex programming

A class of functions called pseudo B-vex and quasi B-vex functions is introduced by relaxing the definitions of B-vex, pseudoconvex, and quasiconvex functions. Similarly, the class of B-invex, pseudo B-invex, and quasi B-invex functions is defined as a generalization of B-vex, pseudo B-vex, and quasi B-vex functions. The sufficient optimality conditions and duality results are obtained for a nonlinear programming problem involving B-vex and B-invex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The second author is grateful to the Faculty of Management, University of Manitoba for the financial support provided for her visit. The authors are thankful to Prof. R. N. Kaul, Department of Mathematics, Delhi University for his constructive criticism of the paper.     

Theoretical investigations on maximal dual feasible functions

Dual feasible functions are used to get valid inequalities and lower bounds for integer linear problems. In this paper, we provide a simpler proof for maximality, and we describe new results concerning the extremality of functions from the literature. Extremal functions are a dominant class of dual feasible functions.     

Semi-B-Preinvex Functions

In this note, a class of functions, called semi-B-preinvex function, which are a generalization of the semipreinvex functions and the B-vex functions, is introduced. Examples are given to show that there exist functions which are semi-B-preinvex functions, but are neither semipreinvex nor B-vex. A property of the semi-B-preinvex functions is obtained.Communicated by F. GiannessiThis research was partially supported by a Science Committee Project, Research Foundation of Chongqing, Grant 8409. The authors are thankful to the referees and Prof. F. Giannessi for suggestions and comments which helped to give the present form to this paper.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on L² traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions.     

Generalization of preinvex and B-vex functions

A class of functions called B-preinvex functions is introduced by relaxing the definitions of preinvex and B-vex functions. Examples are given to show that there exist functions which are B-preinvex but not preinvex or B-vex or quasipreinvex. Some of the properties of B-preinvex functions are obtained.The contribution of this author was made partially at University of Manitoba and partially at St. Lawrence University during the summer of 1990. Support for the visit to St. Lawrence University was made possible under the GTE/SLU Visiting Scholars Grant.     

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.     

Characterizations and Applications of Prequasi-Invex Functions

In this paper, two new types of generalized convex functions are introduced. They are called strictly prequasi-invex functions and semistrictly prequasi-invex functions. Note that prequasi-invexity does not imply semistrict prequasi-invexity. The characterization of prequasi-invex functions is established under the condition of lower semicontinuity, upper semicontinuity, and semistrict prequasi-invexity, respectively. Furthermore, the characterization of semistrictly prequasi-invex functions is also obtained under the condition of prequasi-invexity and lower semicontinuity, respectively. A similar result is also obtained for strictly prequasi-invex functions. It is worth noting that these characterizations reveal various interesting relationships among prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions. Finally, prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions are used in the study of optimization problems.     

On the Gap Functions of Prevariational Inequalities

Gap functions play a crucial role in transforming a variational inequality problem into an optimization problem. Then, methods solving an optimization problem can be exploited for finding a solution of a variational inequality problem. It is known that the so-called prevariational inequality is closely related to some generalized convex functions, such as linear fractional functions. In this paper, gap functions for several kinds of prevariational inequalities are investigated. More specifically, prevariational inequalities, extended prevariational inequalities, and extended weak vector prevariational inequalities are considered. Furthermore, a class of gap functions for inequality constrained prevariational inequalities is investigated via a nonlinear Lagrangian.     

New families of perfect nonlinear polynomial functions

A new class of almost perfect nonlinear (APN) polynomial functions has been recently introduced. We give some generalizations of these functions and deduce new families of perfect nonlinear (PN) functions. We show that these PN functions are CCZ-inequivalent to the known perfect nonlinear functions.     

Functional inequalities involving special functions II

This paper is motivated by an open problem of Anderson, Vamanamurthy, and Vuorinen. We prove a chain of inequalities for hypergeometric functions, generalized and normalized Bessel functions, Kummer functions and for some general power series. Some particular cases and refinements are given.     

P-凸函数及其性质

给出P-凸函数的定义及其性质,将一个有关凸函数的命题推广到P-凸函数情形,并指出赵海清等人关于该命题的证明过程是错误的.     

PROPERTIES OF THE CONVOLUTION WITH PRESTARLIKE FUNCTIONS

In the paper we investigate convolution properties related to the prestarlike functions and various inclusion relationships between defined classes of functions. Interest-ing applications involving the well-known classes of functions defined by linear operators are also considered.     

参数函数和复合函数高阶导数的计算

用函数的链式法则和乘积公式给出了参数函数和复合函数的高阶导数的计算公式．     

Convex composite functions in Banach spaces and the primal lower-nice property

Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are ``integrable' (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower- functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.

     

Inequalities for Humbert functions

This paper is motivated by an open problem of Luke’s theorem. We consider the problem of developing a unified point of view on the theory of inequalities of Humbert functions and of their general ratios are obtained. Some particular cases and refinements are given. Finally, we obtain some important results involving inequalities of Bessel and Whittaker’s functions as applications.