Sufficient Conditions For Convexity Research Articles

Certain classes of meromorphic multivalent functions

In this paper, we introduce and investigate two new classes of meromorphic multivalent functions. Such results as subordination properties, coefficient inequalities, convolution properties and integral representations are proved. Several sufficient conditions for meromorphic multivalent starlikeness and convexity are also derived.

Some simple sufficient conditions for starlikeness and convexity

Let A be the class of analytic functions in the unit disc that are of the form f ( z ) = z + a 2 z 2 + ⋯ . In this work the expressions f ′ ( z ) − ( 1 − γ ) f ( z ) / z and z f ″ ( z ) − γ f ′ ( z ) , 0 < γ ≤ 1 , are studied, as sufficient conditions for starlikeness and convexity.

A convexity problem and a new proof for linearity preserving additions of fuzzy intervals

In this paper, we propose a new sufficient condition for convexity of function. This result provides a simple new proof for linearity preserving additions of fuzzy intervals obtained by the author in 2002, which was a conjecture suggested by Mesiar in 1997.

A class of new trigonometric inequalities and their sharpenings

Some sufficient or necessary conditions for Schur-convexity of a function of two variables F(x,y )=( f (y) − f (x))/(g(y) − g(x)) were considered. These results are applied to establish a class new inequalities in a triangle. In the fourth section we prove two theorems for a kind of symmetric function. These theorems are used to sharpen some of the inequalities and yield two inequalities in the last section. In the paper (5), the author consider necessary and sufficient conditions for convexity of a function xf (x) in terms of some properties of the associated function of two variables F(x,y )=( f (y) − f (x))/(y − x). These results are ap- plied to the theory of the Gamma function. This paper follows the ideas in (5) and discusses the conditions for Schur-convexity and some statements of the function F(x,y )=( f (y) − f (x))/(g(y) − g(x)), and use them to yield some new inequali- ties in a triangle. For example it is shown that in an acuteABC,

Communication and Cooperation in Public Network Situations

This paper focuses on sharing the costs and revenues of maintaining a public network communication structure. Revenues are assumed to be bilateral and communication links are publicly available but costly. It is assumed that agents are located at the vertices of an undirected graph in which the edges represent all possible communication links. We take the approach from cooperative game theory and focus on the corresponding network game in coalitional form which relates any coalition of agents to its highest possible net benefit, i.e., the net benefit corresponding to an optimal operative network. Although finding an optimal network in general is a difficult problem, it is shown that corresponding network games are (totally) balanced. In the proof of this result a specific relaxation, duality and techniques of linear production games with committee control play a role. Sufficient conditions for convexity of network games are derived. Possible extensions of the model and its results are discussed.

Generalized index for Hamiltonian systems with applications

Some classes of linear Hamiltonian equations with periodic coefficients (nondegenerate, strongly stable, completely unstable) are determined by the disposition of the Floquet multipliers. Periodic solutions of nonlinear equations (e.g. elliptic or hyperbolic solutions) are also defined by the multipliers of the corresponding variational equation. In this paper, we consider a general set M of linear Hamiltonian equations with multipliers satisfying some arbitrary conditions and a specific condition on the multiplier lying at some point of the unit circle (all known sets admit such a definition). We show that the set M consists of a finite number of subsets Mi which comprise a countable number of domains within which any two Hamiltonians can be continuously deformed into each other. The corresponding integer index q is expressed through the eigenvalues of some self-adjoint problem. It is shown that this index (and, therefore, the known indices relating to specific sets) increases on increasing the Hamiltonian. Using the obtained results, some known and new sets are studied from the unified point of view. It is shown that for the sets of nondegenerate and completely unstable equations, the domains are directionally convex; for strongly stable equations, necessary and sufficient conditions for directional convexity are found. The results are applied to problems of existence and stability of periodic solutions of nonlinear Hamiltonian equations.

Characterizations of hyperbolically convex regions

Let X be a simply connected and hyperbolic subregion of the complex plane C . A proper subregion Ω of X is called hyperbolically convex in X if for any two points A and B in Ω, the hyperbolic geodesic arc joining A and B in X is always contained in Ω. We establish a number of characterizations of hyperbolically convex regions Ω in X in terms of the relative hyperbolic density ρ Ω ( w ) of the hyperbolic metric of Ω to X, that is the ratio of the hyperbolic metric λ Ω ( w ) | d w | of Ω to the hyperbolic metric λ X ( w ) | d w | of X. Introduction of hyperbolic differential operators on X makes calculations much simpler and gives analogous results to some known characterizations for euclidean or spherical convex regions. The notion of hyperbolic concavity relative to X for real-valued functions on Ω is also given to describe some sufficient conditions for hyperbolic convexity.

Sufficient conditions for convexity in a class of functions arising in telecommunications

Convexity is a key property for a class of functions arising in telecommunications. We derive sufficient conditions for this to hold.

Distributed Control of Heterogeneous Systems

This work considers control design for distributed systems, where the controller is to adopt and preserve the distributed spatial structure of the nominal system. The specific feature of this work is that it does not require the underlying system dynamics to be homogeneous (shift invariant) with respect to spatial or temporal variables. Operator theoretic tools for working with these systems are developed, and lead to sufficient convex conditions for analysis and synthesis with respect to the /spl lscr//sub 2/-induced norm.

Convexity of Images of Convex Sets under Smooth Maps

The article presents sufficient conditions for convexity of images of convex sets under smooth mappings of finite-dimensional spaces of the same dimensions. Applications of the results to nonlinear programming are discussed. Efficient convexity tests are derived for images of Lebesgue sets of convex functions under C1, 1 maps.

Appell polynomials and logarithmic convexity

In this article we give necessary and sufficient conditions for logarithmic convexity of some sequences of Appell polynomials. Then we apply our results to Turan’s type polynomial inequalities. Precise upper and lower bounds for this class of polynomials are also determined and asymptotic behavior of

as well.

A note on fuzzy convexity

In this paper, we introduce and study the concept of semistrictly convex fuzzy sets.We also prove that for the upper semicontinuous case, the class of semistrictly convex fuzzy sets lies between the convex and strictly convex classes. In addition, sufficient conditions for strict convexity are given, and important connections between these convex fuzzy sets are presented.

Convexity of the joint numerical range: topological and differential geometric viewpoints

We investigate the convexity of the joint numerical range of m-tuples of n× n hermitian matrices. The methods come from differential geometry and the differential and algebraic topology. Our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n. Modulo a mild technical assumption, this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the ( m−1)-dimensional unit sphere. The condition is that the eigenvalue has constant multiplicity. We show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges. As a byproduct of our main result, we obtain a new proof of the celebrated Toeplitz–Hausdorff theorem, and of its extension: The numerical range of any triple of hermitian n× n matrices is convex if n>2.

Dynamic convexity for natural thermostatted systems

Natural thermostatted systems are mechanical systems whose Lagrangian is the difference of a kinetic and a potential energy, subjected to the nonholonomic constraint of a constant kinetic energy. When any two points of the configuration space are joined by a thermostatted motion, we say that the system is dynamically convex. A thermostatted charged particle on the plane with a constant electric field is not a dynamically convex system. We prove a general sufficient condition for dynamic convexity, from which whole classes of examples are easily constructed.

Solving generalized transportation problems via pure transportation problems

AbstractThis paper investigates certain issues of coefficient sensitivity in generalized network problems when such problems have small gains or losses. In these instances, it might be computationally advantageous to temporarily ignore these gains or losses and solve the resultant “pure” network problem. Subsequently, the optimal solution to the pure problem could be used to derive the optimal solution to the original generalized network problem. In this paper we focus on generalized transportation problems and consider the following question: Given an optimal solution to the pure transportation problem, under what conditions will the optimal solution to the original generalized transportation problem have the same basic variables? We study special cases of the generalized transportation problem in terms of convexity with respect to a basis. For the special case when all gains or losses are identical, we show that convexity holds. We use this result to determine conditions on the magnitude of the gains or losses such that the optimal solutions to both the generalized transportation problem and the associated pure transportation problem have the same basic variables. For more general cases, we establish sufficient conditions for convexity and feasibility. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 666–685, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10034

Multilinear direct and reverse Stolarsky inequalities

For any nonnegative measurable function f : [0, 1] → R and any a > 0 , let Q (f , a) denote the Stolarsky transform of f , equal to ∫ 1 0 f ( x1/a ) dx . Let Sn stand for the set of all permutations of the set {1, . . . , n} . It is shown that the function (0,∞)n a = (a1, . . . , an) −→ Q (a) := ∑ σ∈Sn n ∏ i=1 Q ( fσ(i), ai ) is Schur-convex if the functions f 1, . . . , f n are nonnegative and nondecreasing and Schur-concave if f 1, . . . , f n are nonnegative and nonincreasing. Necessary and sufficient conditions for the strict Schur convexity and concavity are given. Similar results are obtained for certain “direct” and “reverse” extensions of the Stolarsky transform to measures. Mathematics subject classification (2000): 26D15, 26B25, 26A48.

Some necessary and sufficient conditions for convexity on bounded balanced pseudoconvex domains in

Let ω be a bounded balanced pseudoconvex domain with C 1-plurisubharmonic defining functions in . We give some necessary and sufficient conditions for a locally binoiomorphic mapping from Ω into to be convex.

The analysis of optimization based controllers

Many control techniques employ on-line optimization in the determination of a control policy. We develop a framework which provides sufficient convex conditions, in the form of linear matrix inequalities, for the analysis of constrained quadratic based optimization schemes. These results encompass standard robustness analysis problems for a wide variety of receding horizon control schemes, including those with polytopic and structured uncertainty. A simple example illustrates the methodology.

Convexity ranks in higher dimensions

A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function % is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then % is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists α 2 and are seen to be drastically more complicated than uncountably convex closed subsets of R2.

Another characterization of convexity for set-valued maps

We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative of F which is constructed with the help of an abstract subdifferential notion of Penot . A detailed discussion is devoted to special cases when the contingent, the Fréchet and the Clarke-Rockafellar subdifFerentials Sixe used as this abstract subdifferential.