Abstract convexity of extended real-valued ICR functions

The theory of non-negative increasing and co-radiant (ICR) functions defined on ordered topological vector spaces has been well developed. In this article, we present the theory of extended real-valued ICR functions defined on an ordered topological vector space X. We first give a characterization for non-positive ICR functions and examine abstract convexity of this class of functions. We also investigate polar function and subdifferential of these functions. Finally, we characterize abstract convexity, support set and subdifferential of extended real-valued ICR functions.     

Abstract convexity of radiant functions with applications

In this paper, we investigate abstract convexity of non-positive increasing and radiant (IR) functions over a topological vector space. We characterize the essential results of abstract convexity such as support set, subdifferential and polarity of these functions. We also give some characterizations of a certain kind of polarity and separation property for non-convex radiant and co-radiant sets.     

Abstract convexity:examples and applications

This paper presents a survey of some results from and applications of abstract convexity based on the notions of Minkowski duality, supremal generators, subdifferentials and conjugations. The paper contains very many examples, which are its essential part     

Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by

     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

Second-order subdifferentials and convexity of real-valued functions

In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of C¹ functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given.     

Automatic convexity of rank-1 convex functions

We announce new structural properties of 1-homogeneous rank-1 convex integrands, and discuss some of their consequences.     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Communicated by T.N.T. Goodman     

Generalized convexity and inequalities involving special functions

In this paper, the authors show a relation between the generalized convexity and super- (sub-)multiplicative property, and discuss some generalized convexity and inequalities involving the Gaussian hypergeometric function, the generalized η-distortion function and the generalized Grötzsch function μa(r).     

A convexity property of zeros of Bessel functions

Fork=1, 2,... letj _vk andc _vk be thek-th positive zeros of the Bessel function $$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$ whereY _v (X) is the Bessel function of the second kind. Using the notationj _vκ =C _vk withκ=k?α/π introduced in [3] we show that the functionj _vκ +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j _0κ +j _1κ ? 2κπ > log 8/9, where κ≥0.7070....     

Multivariate convexity preserving interpolation by smooth functions

A set of multivariate data is called strictly convex if there exists a strictly convex interpolant to these data. In this paper we characterize strict convexity of Lagrange and Hermite multivariate data by a simple property and show that for strict convex data and given smoothness requirements there exists a smooth strictly convex interpolant. We also show how to construct a multivariate convex smooth interpolant to scattered data. Partially supported by DGICYT PS93-0310 and by the EC project CHRX-CT94-0522.     

Schur convexity for a class of symmetric functions

The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in Some properties of a class of symmetric functions is answered. As consequences, some inequalities are established by use of the theory of majorization.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation

Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→_sinp?(y)$p\to {\sin }_p?(y)$ and p→_cosp?(y)$p\to {\cos }_p?(y)$ are log-concave on the appropriate intervals while p→_tanp?(y)$p\to {\tan }_p?(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].     

Structural results on convexity relative to cost functions

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen’s inequality for c-convex functions.     

Generalized monotonicity and convexity of non-differentiable functions

The relationships between (strict, strong) convexity of non-differentiable functions and (strict, strong) monotonicity of set-valued mappings, and (strict, strong, sharp) pseudo convexity of non-differentiable functions and (strict, strong) pseudo monotonicity of set-valued mappings, as well as quasi convexity of non-differentiable functions and quasi monotonicity of set-valued mappings are studied in this paper. In addition, the relations between generalized convexity of non-differentiable functions and generalized co-coerciveness of set-valued mappings are also analyzed.