ON UNIVALENT BLOCH FUNCTIONS

Let A = {z: lzl < 1} be the unit disk of complex plane C. For a 6 A, let W.(z) = isbe the M5biu8 transfOrlnation of A and let g(z, a) = log Ieel be the Green's functiou on Awith logarithlllic sillgularity at a E a. FOr 0 < p < oo, tl1e space Q. consists of all fuuctiolls fa11alytic il1 A for w11ichsup / I If'(z)l'(g(z, a))P dA(z) < oo, (1)aea j ja If'(z)l'(g(z, a))P dA(z) < oo, (1)wllere dA(z) is tlle Euclidcan area eIelllent o11 A.Q.-spaces have been i11vestigated by n1any authors (…     

Generalized Invexity and Generalized Invariant Monotonicity

In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.     

Generalized Growth Orders for Polymonogenic Functions and Related Inequalities

In this paper we introduce generalized growth orders and growth types in the sense of Shah and Seremeta in the context of polymonogenic functions. They provide a refinement of previously introduced growth orders in the study of the asymptotic growth behavior of functions satisfying higher dimensional Cauchy–Riemann type equations. We prove a number of insight giving inequalities and manage to extend some basic results that were recently proved for the context of the usual first order Cauchy–Riemann system in Clifford analysis.     

On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.      

On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.     

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.     

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion of G^∞-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.     

Stable generalization of convex functions

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.     

Analytical representation of concave and quasiconcave homogeneous functions

The formula which implements bijection between the class of concave linearly homogeneous functions defined on the nonnegative orthant of an arithmetic space and the simpler class of concave functions defined on the standard (probabilistic) simplex is presented. Two generalizations of this formula for analytical representation of quasiconcave homogeneous function are also proposed. These formulas particularly extend opportunities of modelling production objects and consumption.     

Higher-order conditions for strict efficiency

Luu and Kien (On higher order conditions for strict efficiency, Soochow J. Math. 33 (2007), pp. 17–31), proposed higher-order conditions for strict efficiency of vector optimization problems based on the derivatives introduced in Ginchev (Higher order optimality conditions in nonsmooth optimization, Optimization 51 (2002), pp. 47–72). These derivatives are defined for scalar functions and in their terms necessary and sufficient conditions are obtained a point to be strictly efficient (isolated) minimizer of a given order for quite arbitrary scalar function. Passing to vector functions, Luu and Kien lose the peculiarity that the optimality conditions work with arbitrary functions. In this article, applying the mentioned derivatives for the scalarized problem and restoring the original idea, optimality conditions for strict efficiency of a given order are proposed, which work with quite arbitrary vector functions. It is shown that the results of Luu and Kien are corollaries of the given conditions and generalizations are discussed.     

Multiple Cotangent and Generalized Eta Functions

In this paper we shall construct multiple analogue of the cotangent functions by using the multiple Hurwitz zeta functions and study their properties and special values. In particular, we express the double cotangent functions in terms of generalized eta functions of Berndt and Lewittes.     

What generalizations do students achieve with respect to trigonometric functions in the transition from angles in degrees to real numbers?

The aim of this study is to examine prospective mathematics teachers’ generalizations of trigonometric functions from the unit circle to the Cartesian coordinate system. The researcher developed a test that aimed to reveal students’ generalizations, as well as the possible differences between them. The test was administered to 30 students who were near completion of their university degree program. The findings showed that the students were unable to establish the link between the unit circle and the Cartesian coordinate representation system; and therefore, they were not able to interpret the outputs of trigonometric functions with input of a real number that is not a multiple of π. The researcher also found that the students had developed certain misconceptions regarding the properties of trigonometric functions. To improve the teaching of trigonometric functions an instructional sequence and an alternative definition for trigonometric functions is proposed.     

Notes on L-/M-convex functions and the separation theorems

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000     

Some Further Generalizations of Ostrowski Inequality for Hölder Functions and Functions with Bounded Derivatives

Some generalizations of Ostrowski inequality for Hölder functions and functions with bounded derivatives are given.     

Generalized Hermitean Clifford–Hermite polynomials and the associated wavelet transform

Clifford analysis is a higher‐dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing higher‐dimensional continuous wavelet transforms, the construction of the wavelets being based on generalizations to a higher dimension of classical orthogonal polynomials on the real line. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focusses on so‐called Hermitean monogenic functions, i.e. simultaneous null solutions of two Hermitean Dirac operators. In this Hermitean setting, Clifford–Hermite polynomials and their associated families of wavelet kernels have been constructed starting from a Rodrigues formula involving both Hermitean Dirac operators mentioned. Unfortunately, the property of the so‐called vanishing moments of the corresponding mother wavelets, ensuring that polynomial behaviour in the analyzed signal is filtered out, is only partially satisfied and has to be interpreted with care, the underlying mathematical reason being the fact that the Hermitean Clifford–Hermite polynomials show a too restrictive structure. In this paper, we will remediate this drawback by considering generalized Hermitean Clifford–Hermite polynomials, involving in their definition homogeneous Hermitean monogenic polynomials. The ultimate goal being the construction of new continuous wavelet transforms by means of these polynomials, we first deeply investigate their properties, amongst which are their connection with the traditional Laguerre polynomials, their structure and recurrence relations. Copyright © 2008 John Wiley & Sons, Ltd.     

Generalized Higher-Order Algebraic Differential Equations with Admissible Algebroid Solutions

§ 1.ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓ　ＩｎｔｈｉｓｐａｐｅｒｗｅｕｓｅｔｈｅｓｔａｎｄａｒｄｎｏｔａｔｉｏｎｓｏｆｔｈｅＮｅｖａｎｌｉｎｎａｔｈｅｏｒｙｏｆｍｅｒｏｍｏｒｐｈｉｃｆｕｎｃｔｉｏｎｓｏｒａｌｇｅｂｒｏｉｄｆｕｎｃｔｉｏｎｓ.Ｆｏｒｒｅｆｅｒｅｎｃｅｓｅｅｅ.ｇ.[1 ] .　Ｃｏｎｓｉｄｅｒｔｈｅｇｅｎｅｒａｌｉｚｅｄｈｉｇｈｅｒ ｏｒｄｅｒａｌｇｅｂｒａｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎΩ1(ｚ,ｗ)Ω2 (ｚ,ｗ) =Ｐ(ｚ,ｗ)Ｑ(ｚ,ｗ) , ( 1 )ｗｈｅｒｅΩ1(ｚ,ｗ) =∑…     

半空间中一类次调和函数的增长性质

在Rn的半空间{x∈Rn,xn>0}中,得到了具有Dirichlet数据的Poisson积分在自然的积分收敛条件下满足增长性质u(x)=o(|x|),这里|x|→∞,这一性质对于半空间中满足一定条件的次调和函数仍然成立.该结果把复平面C中解析函数的增长性质推广到了n-维Euclidean半空间,并且推广了n-维Euclidean半空间中某些经典的结果.     

Generalized solutions of the Vlasov-Poisson system with singular data

We study spherically symmetric solutions of the Vlasov-Poisson system in the context of algebras of generalized functions. This allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system. The proof of unique solvability in our approach depends on new stability properties of the system with respect to perturbations.     

Sobolev Embedding Theorems and Generalizations for Functions on a Metric Measure Space

Considering the metric case, we define an analog of the Sobolev space of functions with generalized derivatives of order greater than 1. The space of functions with fractional generalized derivatives is also treated. We prove generalizations of the Sobolev embedding theorems and Gagliardo–Nirenberg interpolation inequalities to the metric case.