THE FEKETE-SZEG PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO THE KOEBE FUNCTION

An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.

Higher Order Strongly Biconvex Functions and Biequilibrium Problems

In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.

Differentiability of Convex Functions in Locally Convex Spaces(Ⅱ)──Frechet

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SOME INEQUALITIES OF HERMITE-HADAMARD TYPE FOR s-CONVEX FUNCTIONS

In this paper several inequalities of the left-hand side of Hermite-Hadamard’s inequality are obtained for s-convex functions.

DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES

Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.

Inequalities for Tφ-convex functions

In this paper, the Tφ-convex functions were introduced as a generalizations of convex functions. Then the characteristics of the Tφ-convex functions were discussed. Furthermore, some new inequalities for the Tφ-convex functions were derived.

Several Hermite-Hadamard Type Inequalities for Harmonically Convex Functions in the Second Sense with Applications

In this paper, we first introduce the concept ＂harmonically convex functions＂ in the second sense and establish several Hermite-Hadamard type inequalities for harmonically convex functions in the second sense. Finally, some applications to special mean are shown.

A Class of Schur Convex Functions and Several Geometric Inequalities

Schur convexity, Schur geometrical convexity and Schur harmonic convexityof a class of symmetric functions are investigated. As consequences some knowninequalities are generalized. In addition, a class of geometric inequalities involvingn-dimensional simplex in n-dimensional Euclidean space En and several matrix inequalitiesare established to show the applications of our results.

Some Hermite-Hadamard Type Inequalities for Differentiable Co-Ordinated Convex Functions and Applications

In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality in two variables. In addition, six other inequalities are derived from it for some refinements. Finally, this paper shows some examples that these inequalities are able to be applied to some special means.

Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies

In this paper,by making use of Divergence theorem for multiple integrals,we establish some integral inequalities for Schur convex functions defined on bodies B⊂R^(n)that are symmetric,convex and have nonempty interiors.Examples for three dimensional balls are also provided.

Lipschitz-continuity of Spherically Convex Functions

In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functions is equi-Lipschitzian on each closed spherically convex subset contained in the relative interior of their common domain,and from which a powerful result is derived:the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence on each closed spherically convex subset contained in the relative interior of their common domain.

Fekete-Szego problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Given α∈[0, 1], let hα（z） ：= z/（1 - αz）, z ∈ D ：= {z ∈ C： ｜z｜ 〈 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to hα if there exists δ ∈ （-π/2, π/2） such that Re{e^iδ zf′（z）/hα（z）} 〉 0, z ∈ D. For the class l（hα） of all close-to-convex functions with respect to hα, the Fekete-Szego problem is studied.

Distributed Stochastic Optimization with Compression for Non-Strongly Convex Objectives

We are investigating the distributed optimization problem,where a network of nodes works together to minimize a global objective that is a finite sum of their stored local functions.Since nodes exchange optimization parameters through the wireless network,large-scale training models can create communication bottlenecks,resulting in slower training times.To address this issue,CHOCO-SGD was proposed,which allows compressing information with arbitrary precision without reducing the convergence rate for strongly convex objective functions.Nevertheless,most convex functions are not strongly convex(such as logistic regression or Lasso),which raises the question of whether this algorithm can be applied to non-strongly convex functions.In this paper,we provide the first theoretical analysis of the convergence rate of CHOCO-SGD on non-strongly convex objectives.We derive a sufficient condition,which limits the fidelity of compression,to guarantee convergence.Moreover,our analysis demonstrates that within the fidelity threshold,this algorithm can significantly reduce transmission burden while maintaining the same convergence rate order as its no-compression equivalent.Numerical experiments further validate the theoretical findings by demonstrating that CHOCO-SGD improves communication efficiency and keeps the same convergence rate order simultaneously.And experiments also show that the algorithm fails to converge with low compression fidelity and in time-varying topologies.Overall,our study offers valuable insights into the potential applicability of CHOCO-SGD for non-strongly convex objectives.Additionally,we provide practical guidelines for researchers seeking to utilize this algorithm in real-world scenarios.

Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

Two Inequalities for Convex Functions

Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).

Relations between Lipschitz functions and convex functions

We discuss the relationship between Lipschitz functions and convex functions.By these relations, we give a sufficient condition for the set of points where Lipschitz functions on a Hilbert space is Frechet differentiable to be residual.

A fuzzy imperialistic competitive algorithm for optimizing convex functions

Purpose–The purpose of this paper is to describe imperialist competitive algorithm(ICA),a novel socio-politically inspired optimization strategy for proposing a fuzzy variant of this algorithm.ICA is a meta-heuristic algorithm for dealing with different optimization tasks.The basis of the algorithm is inspired by imperialistic competition.It attempts to present the social policy of imperialisms(referred to empires)to control more countries(referred to colonies)and use their sources.If one empire loses its power,among the others making a competition to take possession of it.Design/methodology/approach–In fuzzy imperialist competitive algorithm(FICA),the colonies have a degree of belonging to their imperialists and the top imperialist,as in fuzzy logic,rather than belonging completely to just one empire therefore the colonies move toward the superior empire and their relevant empires.Simultaneously for balancing the exploration and exploitation abilities of the ICA.The algorithms are used for optimization have shortcoming to deal with accuracy rate and local optimum trap and they need complex tuning procedures.FICA is proposed a way for optimizing convex function with high accuracy and avoiding to trap in local optima rather than using original ICA algorithm by implementing fuzzy logic on it.Findings–Therefore several solution procedures,including ICA,FICA,genetic algorithm,particle swarm optimization,tabu search and simulated annealing optimization algorithm are considered.Finally numerical experiments are carried out to evaluate the effectiveness of models as well as solution procedures.Test results present the suitability of the proposed fuzzy ICA for convex functions with little fluctuations.Originality/value–The proposed evolutionary algorithm,FICA,can be used in diverse areas of optimization problems where convex functions properties are appeared including,industrial planning,resource allocation,scheduling,decision making,pattern recognition and machine learning(optimization techniques;fuzzy logic;convex functions).

Convex Functions, Subdifferentiability and Renormings

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.

CERTAIN SUFFICIENT CONDITIONS FOR STARLIKENESS AND CONVEXITY OF MEROMORPHICALLY MULTIVALENT FUNCTIONS

In this paper we derive certain sufficient conditions for starlikeness and convexity of order α of meromorphically multivalent functions in the punctured unit disk.

On order-preserving and order-reversing mappings defined on cones of convex functions

In this paper,we first show that for a Banach space X,there is a fully order-reversing mapping T from conv(X)(the cone of all the extended real-valued lower semicontinuous proper convex functions defined on X)onto itself if and only if X is reflexive and linearly isomorphic to its dual X^(*).Then we further prove the following generalized Artstein-Avidan-Milman representation theorem:For every fully order-reversing mapping T:conv(X)→conv(X),there exist a linear isomorphism U:X→X^(*),x_(0)^(*),φ_(0)∈X^(*),α>0 and r_0∈R so that(Tf)(x)=α(Ff)(Ux+x_(0)^(*))+<φ_(0),x>+r_(0),■x∈X where T:conv(X)→conv(X^(*))is the Fenchel transform.Hence,these resolve two open questions.We also show several representation theorems of fully order-preserving mappings defined on certain cones of convex functions.For example,for every fully order-preserving mapping S:semn(X)→semn(X),there is a linear isomorphism U:X→X so that(Sf)(x)=f(Ux),■f∈semn(X),x∈X where semn(X)is the cone of all the lower semicontinuous seminorms on X.