Convex upper and lower bounds for present value functions

In this paper we present an efficient methodology for approximating the distribution function of the net present value of a series of cash‐flows, when discounting is presented by a stochastic differential equation as in the Vasicek model and in the Ho–Lee model. Upper and lower bounds in convexity order are obtained. The high accuracy of the method is illustrated for cash‐flows for which no analytical results are available. Copyright © 2001 John Wiley & Sons, Ltd.     

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.     

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C⁴‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.     

One class of separable optimization problems: solution method,application

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ? ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.     

Monotonicity properties and functional inequalities for the Volterra and incomplete Volterra functions

In this paper we prove some monotonicity, log–convexity and log–concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Turán type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means.     

半模糊凸模糊映射

In this paper, a new class of fuzzy mappings called semistrictly convex fuzzy mappings is introduced and we present some properties of this kind of fuzzy mappings. In particular, we prove that a local minimum of a semistrictly convex fuzzy mapping is also a global minimum. We also discuss the relations among convexity, strict convexity and semistrict convexity of fuzzy mapping, and give several sufficient conditions for convexity and semistrict convexity.     

Korovkin type theorems and approximate Hermite–Hadamard inequalities

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate convexity property. The failure of such an implication with constant error term shows that functional error terms should be considered for the inequalities and convexity properties in question. The key for the proof of the main result is a Korovkin type theorem which enables us to deduce the approximate convexity property from the approximate lower Hermite–Hadamard type inequality via an iteration process.     

Multiobjective programming under d-invexity

In this paper, a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable. By considering the concept of Pareto optimal solution and substituting d-invexity for convexity, the Fritz John type and Karush–Kuhn–Tucker type necessary optimality conditions and duality in the sense of Mond–Weir and Wolfe for nondifferentiable multiobjective programming are given.     

New geometric properties of Banach spaces

In this paper, we introduce new geometric properties as generalizations of p‐uniform smoothness and q‐uniform convexity of Banach spaces. Furthermore, using generalized Beckner's inequality, we characterize the properties in terms of norm inequalities. As an application, we consider the duality relation.     

On Injectivity of a Class of Monotone Operators with Some Univalency Consequences

In this paper, we provide sufficient conditions that ensure the convexity of the inverse images of an operator, monotone in some sense. Further, conditions that ensure the monotonicity, respectively the local injectivity of an operator, are also obtained. Combining the conditions that provide the local injectivity, respectively the convexity of the inverse images of an operator, we are able to obtain some global injectivity results. As applications, some new analytical conditions that assure the injectivity, respectively univalency of a complex function of one complex variable are obtained. We also show that some classical results, such as Alexander–Noshiro–Warschawski and Wolff theorem or Mocanu theorem, are easy consequences of our results.     

An Univalence Criterion for Analytic Functions Defined in Type $$\varphi $$ Convex Domains

In the present paper we introduce a new characterization of the convexity of a planar domain, based on the convexity constant K(D) of a domain \(D\subset \mathbb {C}\). We show that in the class of simply connected planar domains, \(K(D) =1\) characterizes the convexity of the domain D. Using the convexity constant of a domain, we derive a sufficient condition for the univalence of an analytic function defined in a type \(\varphi \) convex domain, similar to the one obtained by Reade (Math Soc Jpn 10:255–259, 1958), but involving the modulus instead of the argument of the derivative of the function. As a corollary we obtain the well-known Ozaki–Nunokawa–Krzyz univalence criterion, and we also show that our condition is sharp.     

Optimization of the difference of topical functions

In this paper, we first obtain some properties of topical (increasing and plus-homogeneous) functions in the framework of abstract convexity. Next, we use the Toland–Singer formula to characterize the dual problem for the difference of two topical functions. Finally, we present necessary and sufficient conditions for the global minimum of the difference of two strictly topical functions.     

Some mathematical properties of a DEA model for the joint determination of efficiencies

This paper explores the Kuhn–Tucker conditions and convexity issues in a non-linear DEA model for the joint determination of efficiencies developed by Mar Molinero. It is shown that the usual convexity conditions that apply to Linear Programming problems are satisfied in this case. First order Kuhn–Tucker conditions are derived and interpreted. Estimation strategies are suggested. Some empirical work is reported.     

Existence of solutions to an anisotropic phase‐field model

We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

On Linear Functionals and Hahn-Banach Theorems for Hyperbolic and Bicomplex Modules

We consider modules over the commutative rings of hyperbolic and bicomplex numbers. In both cases they are endowed with norms which take values in non–negative hyperbolic numbers. The exact analogues of the classical versions of the Hahn–Banach theorem are proved together with some of their consequences. Linear functionals on these modules are studied and their relations with the corresponding hyperplanes are established. Finally, we introduce the notion of hyperbolic convexity for hyperbolic modules (in analogy with real, not complex, convexity) and establish its relation with hyperplanes.     

Norm Inequalities Related to the Heron and Heinz Means

In this article, we present several inequalities treating operator means and the Cauchy–Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy–Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.     

On the properties of a class of log-biharmonic functions

The aim of this work is to introduce and investigate a new class of log-biharmonic functions. Certain geometrically motivated properties and results concerning starlikeness, convexity and univalence of elements within this class versus the corresponding harmonic functions are obtained and discussed. In particular, we consider the Goodman–Saff conjecture and prove that the conjecture is true for the logarithms of functions belonging to this class.     

Revising an iterative algorithm for finding a common solution of a system of variational inequalities for two inverse strongly accretive operators

In this paper, we give a short and simple proof of the recent result of Katchang and Kumam (Positivity 15:281–295, 2011). In our proof, we do not assume the uniform convexity of a space.     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

Invertible harmonic mappings of the unit disk onto Dini smooth Jordan domains

In this paper we extend Radó–Kneser–Choquet theorem for the mappings with weak homeomorphic Lipschitz boundary function and Dini's smooth boundary but without restriction on the convexity of the image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Radó–Kneser–Choquet theorem by Alessandrini and Nesi [1] and is used the approximation principle.