Class Of Convex Research Articles

On a generalization of classes of convex in the direction and typically real functions

On a generalization of classes of convex in the direction and typically real functions

Compact Extended Formulations for Low-Rank Functions with Indicator Variables

We study the mixed-integer epigraph of a special class of convex functions with nonconvex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions. Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems. Funding: This work was supported by the National Science Foundation Division of Computing and Communication Foundations [Grant 2006762].

On discrete inequalities for some classes of sequences

Abstract For a given sequence a = ( a 1 , … , a n ) ∈ R n a=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n} , our aim is to obtain an estimate of E n ≔ a 1 + a n 2 − 1 n ∑ i = 1 n a i {E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right| . Several classes of sequences are studied. For each class, an estimate of E n {E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

Global Stabilization of a Bounded Controlled Lorenz System

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball [Formula: see text] given by the level set of a certain Lyapunov function, [Formula: see text]. However, since the minimum point of [Formula: see text] does not coincide with any rest point of Lorenz system, we apply a modified solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF [Formula: see text] for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside [Formula: see text], so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside [Formula: see text] that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Generalized n-Polynomial p-Convexity and Related Inequalities

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p=−1, our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields.

Coefficient Bounds of Bi-Univalent Function Involving Pseudo-Starlikeness Associatedwith Sigmoid Function Defined by Salagean Operator via Chebyshev Polynomial

Some special classes of univalent functions play an important role in geometric function theory because of their geometric properties. Many of such classes have been introduced and studied; some became well known, for example, the classes of convex, starlike, close to convex, strongly convex and strongly starlike functions. Previous studies by Awolere and Oladipupo (2018) now served as motivation and background to investigate certain classes of analytic, univalent and bi-univalent functions in terms of their coefficient bounds involving salagean and sigmoid functions via Chebyshev poynomial. The classes 퐻푛(휆,훽,훾(푠),휙(푧,푡))are newly established classes for which coefficient bounds will be determined. The aim of the present work is to investigate coefficient bound for class 퐻푛(휆,훽,훾(푠),휙(푧,푡))of pseudo-starlikeness associated with sigmoid functions defined by Salagean operator via Chebyshev polynomial, Fekete-szego problem will also be established and the Hankel of the function will be determined

Some inequalities of the Hermite-Hadamard type for two kinds of convex functions

In this paper, we obtain new inequalities of the Hermite-Hadamard type, in two different classes of convex dominated functions. Several known results from the literature are obtained as particular cases of our more general perspective.

K-Fractional Ostrowski Type Inequalities via (s, r)−Convex

We introduce the generalized class named it the class of (s,r)−convex in mixed kind, this class includes s−convex in 1st and 2nd kind, P−convex, quasi convex and the class of ordinary convex. Also, we state the generalization of the classical Ostrowski inequality via k−fractional integrals, which is obtained for functions whose first derivative in absolute values is (s,r)−convex in mixed kind. Moreover, we establish some Ostrowski type inequalities via k−fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s,r)−convex in mixed kind by using different techniques including Hölder’s inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means are given.

Sufficiency criteria for a class of convex functions connected with tangent function

<abstract><p>The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.</p></abstract>

Study of Second Order Hankel Determinants for Convex Functions with Symmetric Points Related with Exponential Function

This work aims to introduce and study a class of convex functions with symmetric points associated with exponential function. The problems under consideration are the first four sharp coefficient bounds, Fekete-Szegö functional, second Hankel determinant. It also includes similar investigations for inverse and logarithmic functions of convex functions with symmetric points associated with exponential function.

Note on minimization of quasi M♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{\ atural }$$\\end{document}-convex functions

For a class of discrete quasi convex functions called semi-strictly quasi M♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{\ atural }$$\\end{document}-convex functions, we investigate fundamental issues relating to minimization, such as optimality condition by local optimality, minimizer cut property, geodesic property, and proximity property. Emphasis is put on comparisons with (usual) M♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{\ atural }$$\\end{document}-convex functions. The same optimality condition and a weaker form of the minimizer cut property hold for semi-strictly quasi M♮\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{\ atural }$$\\end{document}-convex functions, while geodesic property and proximity property fail.

Integral Operators Applied to Classes of Convex and Close-to-Convex Meromorphic p-Valent Functions

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given.

On Hermite-Hadamard-Type Inequalities for Subharmonic Functions Over Circular Ring Domains

In this note, we study the so-called Hermite-Hadamard inequality for the class of subharmonic functions. We first prove an inequality of this type for subharmonic functions over circular ring domains. Next, a new Hermite-Hadamard-type inequality over a disk is deduced. Moreover, we introduce the class of subharmonic functions on the coordinates, which includes the class of convex functions on the coordinates, and establish several new integral inequalities for this class of functions over various product domains: product of disks, product of circular rings and product of a disk and a circular ring.

K-fractional Ostrowski type inequalities via (s, r)-conex

We are introducing very first time a generalized class named it the class of (s, r)-convex in mixed kind, this class includes s-convex in 1st and 2nd kind, P-convex, quasi-convex, and the class of ordinary convex. Also, we would like to state the generalization of the classical Ostrowski inequality via k-fractional integrals, which is obtained for functions whose first derivative in absolute values is (s, r)-convex in mixed kind. Moreover, we establish some Ostrowski type inequalities via k-fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are (s, r)-convex in mixed kind by using different techniques including Holder's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, some applications in terms of special means would also be given.

Risk-averse stochastic optimal control: An efficiently computable statistical upper bound

In this paper, we discuss an application of the Stochastic Dual Dynamic Programming (SDDP) type algorithm to nested risk-averse formulations of Stochastic Optimal Control (SOC) problems. We propose a construction of a statistical upper bound for the optimal value of risk-averse SOC problems. This outlines an approach to a solution of a long standing problem in that area of research. The bound holds for a large class of convex and monotone conditional risk mappings. Finally, we show the validity of the statistical upper bound to solve a real-life stochastic hydro-thermal planning problem.

Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation

<abstract><p>There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-$ h $-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard ($ \mathcal{H.H} $) inequality along with a Jensen-type inequality for harmonically $ h $-Godunova-Levin interval-valued functions (GL-$ \mathcal{IVFS} $). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.</p></abstract>

The second and third Hankel determinants for starlike and convex functions associated with Three-Leaf function

In this paper, we give sharp bounds of the Hankel determinant H2(3)(f) for the coefficients of functions in the class of starlike functions related to a domain that is like three leaves. We also give sharp bounds for the Hankel determinants H3(1)(f) and H2(3)(f) for the coefficients of functions in the class of convex functions related to the three-leaf-like domain.

COEFFICIENT INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS APPROACHING TO CLASS OF CONVEX FUNCTIONS IN THE LIMIT FORM AND CLASS OF STARLIKE FUNCTIONS DIRECTLY

We introduce a class of analytic functions and obtain sharp upper bounds of the functional | a3 - μa22 | for the analytic function ƒ(z) = z + ∑∞n=2 anzn, |z| < 1 belonging to this class with special character that it tends to the class of convex functions as α -> π /2

Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains

The object of the paper is to find complete systems of inequalities relating the perimeter [Formula: see text], the area [Formula: see text] and the Cheeger constant [Formula: see text] of planar sets. To do so, we study the so-called Blaschke–Santaló diagram of the triplet [Formula: see text] for different classes of domains: simply connected sets, convex sets and convex polygons with at most [Formula: see text] sides. We completely determine the diagram in the latter cases except for the class of convex [Formula: see text]-gons when [Formula: see text] is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.