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,...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.展开更多
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 n...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 <i>k</i>-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 <i>k</i>-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.展开更多
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...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.展开更多
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-con...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.展开更多
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. Final...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.展开更多
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 in...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.展开更多
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 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.展开更多
In the paper,a class of functions with bounded turnings involving cardioid domain,are studied in the region of the unit disc.The bounds of|a_(5)|,|a_(6)|,|a_(7)|and the fourth Hankel determinant are obtained,which are...In the paper,a class of functions with bounded turnings involving cardioid domain,are studied in the region of the unit disc.The bounds of|a_(5)|,|a_(6)|,|a_(7)|and the fourth Hankel determinant are obtained,which are more accurate than those obtained by Srivastava.展开更多
For the new subclass B of the bi-univalent functions constructed with the help of the(u,v)-Chebyshev polynomials of the second type,we get estimates for the first two initial coefficients and upper bounds of the Feket...For the new subclass B of the bi-univalent functions constructed with the help of the(u,v)-Chebyshev polynomials of the second type,we get estimates for the first two initial coefficients and upper bounds of the Fekete-Szeg o functional.展开更多
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）...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.展开更多
A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most resu...A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most results for .E-convex functions and E-convex programming in [1] are not true by six counter examples.展开更多
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 ...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).展开更多
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 differenti...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 differentiate to be residual.展开更多
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...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).展开更多
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 interior...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.展开更多
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 con...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.展开更多
This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of c...This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.展开更多
文摘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.
文摘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 <i>k</i>-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 <i>k</i>-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.
文摘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.
基金Project supported by the National Natural Science Foundation of China（Grant No.10271071）
文摘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.
基金The Doctoral Programs Foundation(20113401110009)of Education Ministry of ChinaNatural Science Research Project(2012kj11)of Hefei Normal University+1 种基金Universities Natural Science Foundation(KJ2013A220)of Anhui ProvinceResearch Project of Graduates Innovation Fund(2014yjs02)
文摘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.
基金The Doctoral Programs Foundation(20113401110009) of Education Ministry of Chinathe Natural Science Research Project(2012kj11) of Hefei Normal Universitythe NSF(KJ2013A220) of Anhui Province
文摘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.
文摘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.
基金Supported by the Natural Science Foundation of Anhui Provincial Department of Education(Grant Nos.KJ2020A 0993KJ2020ZD74)+2 种基金the High-Level Talent Research Start-Up Project(Grant No.DC2300000286)the Foundation of Guangzhou Civil Aviation College(Grant Nos.22X041824X4412).
文摘In the paper,a class of functions with bounded turnings involving cardioid domain,are studied in the region of the unit disc.The bounds of|a_(5)|,|a_(6)|,|a_(7)|and the fourth Hankel determinant are obtained,which are more accurate than those obtained by Srivastava.
文摘For the new subclass B of the bi-univalent functions constructed with the help of the(u,v)-Chebyshev polynomials of the second type,we get estimates for the first two initial coefficients and upper bounds of the Fekete-Szeg o functional.
文摘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.
基金National Natural Science Foundation of China(10261001)Science Foundation of Guangxi(0236001)
文摘A class of functions and a sort of nonlinear programming called respectively E-convex functions and E-convex programming were presented and studied recently by Youness in [1], In this paper, we point out the most results for .E-convex functions and E-convex programming in [1] are not true by six counter examples.
基金supported by the National Natural Science Foundation of China(No.10071016)
文摘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).
文摘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 differentiate to be residual.
文摘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).
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China (No.10471114)the Pujian Provincial Natural Science Foundation of China (No.F00021)the Tianyuan Foundation of Mathematics (No,A0324618).
文摘This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.
