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In this paper, we create a new subclass of convex functions given with tangent functions applying the combination of Babalola operators and Binomial series. Moreover, we obtain several important geometric results, including sharp coefficient bounds, sharp Fekete–Szego inequality, Kruskal inequality, and growth and distortion estimates. Furthermore, for functions with logarithmic coefficients, we compute sharp Fekete–Szego inequality and sharp coefficient bounds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new f-divergence measures that we call t-asymmetric/symmetric divergence measure and integral divergence measure, establish their joint convexity and provide some inequalities that connect these f-divergences to the classical one introduced by Csiszar in 1963. Applications for the dichotomy class of convex functions are provided as well. [ABSTRACT FROM AUTHOR]

This work presents new versions of the Hermite-Hadamard Inequality, for (m-F)-convex functions, defined on fractal sets R ς (0 < ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized Hölder-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature. 2020 Mathematics Subject Classification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of quasi-point-separable topological vector spaces, which has the following important properties: (1) in general, the conditions for a topological vector space to be quasi-point-separable is not difficult to verify; (2) the class of quasi-point-separable topological vector spaces is large and includes locally convex topological vector spaces and pseudonorm adjoint topological vector spaces as special cases; (3) every quasi-point-separable Housdorrf topological vector space has the fixed point property (that is, every continuous self-mapping on any given nonempty closed and convex subset has a fixed point), which is the result of the main theorem of this paper. Finally, we provide some concrete examples of quasi-point-separable topological vector spaces, which are not locally convex. [ABSTRACT FROM AUTHOR]

In this paper, a quasi-linear viscoelastic Kirchhoff plate equation with logarithmic source and time delay involving free boundary conditions in a bounded domain is considered. The local existence and global existence are proved, respectively. Under the assumptions on a more general type of relaxation functions and suitable conditions on the coefficients between damping term and delay term, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. As the considered assumption here on the kernel is more general than earlier papers, our result improves and generalizes earlier result in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the relationship between r -convexity and Schur-convexity is investigated first. Moreover, the r -convexity and Schur-convexity of a class of functions are studied. As applications, some new inequalities on Minkowski's inequality are established. [ABSTRACT FROM AUTHOR]

In the paper, the authors correct the paper "Mevlüt Tunç, On some integral inequalities for s-geometrically convex functions and their applications, Int. J. Open Problems Comput. Math. 6 (2013), no. 1, 53-63; available online at http://dx.doi.org/10.12816/0006155." [ABSTRACT FROM AUTHOR]

The paper deals with a significant extension of the recently proposed class of relatively strongly convex optimization problems in spaces of large dimension. In the present paper, we introduce an analog of the concept of relative strong convexity for variational inequalities (relative strong monotonicity) and study estimates for the rate of convergence of some numerical first-order methods for problems of this type. The paper discusses two classes of variational inequalities depending on the conditions related to the smoothness of the operator. The first of these classes of problems contains relatively bounded operators, and the second, operators with an analog of the Lipschitz condition (known as relative smoothness). For variational inequalities with relatively bounded and relatively strongly monotone operators, a version of the subgradient method is studied and an optimal estimate for the rate of convergence is justified. For problems with relatively smooth and relatively strongly monotone operators, we prove the linear rate of convergence of an algorithm with a special organization of the restart procedure of a mirror prox method for variational inequalities with monotone operators. [ABSTRACT FROM AUTHOR]

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. [ABSTRACT FROM AUTHOR]

Subgradient methods are frequently used for optimization problems. However, subgradient techniques are characterized by slow convergence for minimizing ravine convex functions. To accelerate subgradient methods, special linear non-orthogonal transformations of the original space are used. This paper provides an overview of these transformations based on Shor's original idea. Two one-rank linear transformations of Euclidean space are considered. These simple transformations form the basis of variable metric methods for convex minimization that have a natural geometric interpretation in the transformed space. Along with the space transformation, a search direction and a corresponding step size must be defined. Subgradient Fejer-type methods are analyzed to minimize convex functions, and Polyak step size is used for problems with a known optimal objective value. Convergence theorems are provided together with the results of numerical experiments. Directions for future research are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study some characterizations of the class of weakly subdifferentiable functions and formulate optimality conditions for nonconvex mathematical programming problems described by the class of weakly subdifferentiable functions in real normed spaces. The necessary and sufficient optimality conditions for a nonconvex scalar function with a global minimum/or a global maximum at a given vector via the weak subdifferentials and augmented normal cones are established. Additionally, the necessary and sufficient optimality conditions for a nonconvex vector function with a weakly efficient solution/or an efficient solution at a given vector via the augmented weak subdifferentials and normal cones are presented too. Finally, our optimality conditions are used to derive the necessary optimality conditions for nonsmooth nonconvex mathematical programming problems with set, inequality, and equality constraints. [ABSTRACT FROM AUTHOR]

In this paper, we study the quantitative stability of an optimal control problem with respect to parametric perturbations. We essentially obtain two equivalent conclusions for the stability of this problem by using two independent methods. The first one makes recourse to standard computations based on the famous Gronwall Lemma while our second method employees rather stability of fixed points trough the celebrated Lim's Lemma for which we construct a suitable contracting set-valued mapping over a larger functional space than the one of continuous functions adopted in the close previous works. The second method allows us to introduce a further concept of approximate solutions regarded as approximate values of the optimal control for which we prove similar stability properties as in the case of exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the ravine of real-valued functions is extended from the finite-dimensional setting to an infinite-dimensional setting. Ravines of quadratic functions are studied in detail. The obtained results solve a problem raised by Professor Joachim Gwinner. In addition, it is proved that a weakly continuous real-valued convex function defined on a reflexive Banach space cannot have any ravine along the null subspace. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of the optimality and scalarization for approximate solutions to a Constrained Vector Equilibrium Problem (CVEP). The optimality conditions are given in terms of Michel-Penot subdifferentials, and the scalarization theorems are proposed via a strongly monotone cone convex function. We firstly establish a necessary condition for an approximate quasi weakly efficient solution to problem (CVEP). Then, a sufficient condition for approximate quasi Benson proper efficient solutions to problem (CVEP) is examined under the newly introduced generalized convexity assumptions. Finally, by using the properties of Bishop-Phelps cone, we present the scalarization theorems for approximate quasi weakly (Benson proper) efficient solutions. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds of the Hankel determinants H2,1(f) and H2,2(f) for the classes S*(λ,n), where f is analytic in the open unit disk Δ = {z ∈ C: |z| < 1g and normalized so that f(z) = z+a2z²+a3z³+... [ABSTRACT FROM AUTHOR]

In early works, we presented a new economical and effective method for finding the global optimum of a function of many variables, which was conditionally called the auxiliary function method. The essence of the method is that a multi-extremal and multivariable objective function is transformed into a convex function $g_m(F, \alpha)$ of one variable, which is the Lebesgue integral over a compact where the objective function is considered: $g_m(F, \alpha)=\int_E[|F(x)-\alpha|-F(x)+\alpha]^m d \mu$, $m \in N$. The function $g_m(F, \alpha)$ was called the auxiliary function. In early works, the properties of the auxiliary function and the algorithm of the new method were studied, the convergence of the method was proven, and computational experiments were carried out with multiextremal functions in three-dimensional space. Based on these results and in order to demonstrate the advantages of using the auxiliary function method, this paper considers the problem of finding global minima of objective functions in a four-dimensional space constructed on the basis of hyperbolic and exponential potentials and conducts a comparative analysis of the results obtained. In this work, as a result of completed computational experiments on test functions in three-dimensional and four-dimensional space, where auxiliary functions with different values of the degree $m \in N$ were expanded, important conclusions were obtained and proven. As a result, the change in the auxiliary function depending on its degree m is clearly shown. This result provides even more opportunities to improve the efficiency of the constructed method. Next, you can set up first- and second-order methods to find the "oldest" zero auxiliary function. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

Stochastic optimization has experienced significant growth in recent decades, with the increasing prevalence of variance reduction techniques in stochastic optimization algorithms to enhance computational efficiency. In this paper, we introduce two projection-free stochastic approximation algorithms for maximizing diminishing return (DR) submodular functions over convex constraints, building upon the Stochastic Path Integrated Differential EstimatoR (SPIDER) and its variants. Firstly, we present a SPIDER Continuous Greedy (SPIDER-CG) algorithm for the monotone case that guarantees a (1 - e - 1) OPT - ε approximation after O (ε - 1) iterations and O (ε - 2) stochastic gradient computations under the mean-squared smoothness assumption. For the non-monotone case, we develop a SPIDER Frank–Wolfe (SPIDER-FW) algorithm that guarantees a 1 4 (1 - min x ∈ C ‖ x ‖ ∞) OPT - ε approximation with O (ε - 1) iterations and O (ε - 2) stochastic gradient estimates. To address the practical challenge associated with a large number of samples per iteration, we introduce a modified gradient estimator based on SPIDER, leading to a Hybrid SPIDER-FW (Hybrid SPIDER-CG) algorithm, which achieves the same approximation guarantee as SPIDER-FW (SPIDER-CG) algorithm with only O (1) samples per iteration. Numerical experiments on both simulated and real data demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of strongly generalized convex functions which is called as strongly η-convex stochastic processes. We prove the Hermite-Hadamard, Ostrowski type inequality, and obtain some important inequalities for above processes. Some previous results are special cases of the results obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]

A popular approach to minimizing a finite sum of smooth convex functions is stochastic gradient descent (SGD) and its variants. Fundamental research questions associated with SGD include (i) how to find a lower bound on the number of times that the gradient oracle of each individual function must be assessed in order to find an e-minimizer of the overall ob jective; (ii) how to design algorithms which guarantee finding an e-minimizer of the overall ob jective in expectation no more than a certain number of times (in terms of 1/\epsilon) that the gradient oracle of each function needs to be assessed (i.e., upper bound). If these two bounds are at the same order of magnitude, then the algorithms may be called optimal. Most existing results along this line of research typically assume that the functions in the ob jective share the same condition number. In this paper, the first model we study is the problem of minimizing the sum of finitely many strongly convex functions whose condition numbers are all different. We propose an SGD-based method for this model and show that it is optimal in gradient computations, up to a logarithmic factor. We then consider a constrained separate block optimization model and present lower and upper bounds for its gradient computation complexity. Next, we propose solving the Fenchel dual of the constrained block optimization model via generalized SSNM, which we introduce earlier, and show that it yields a lower iteration complexity than solving the original model by the ADMM-type approach. Finally, we extend the analysis to the general composite convex optimization model and obtain gradient-computation complexity results under certain conditions. [ABSTRACT FROM AUTHOR]

We define and look into the fundamental features of X-log convex, X-log quasi convex, X-log semi-strictly quasi convex, and X-log-pseudo convex functions. This paper provides several examples and arguments in favour of the concept. Also, we develope another X-log convex function as the sum of an X-log convex and X-log affine convex function. Additionally, we define local and global log-minimum. Under one condition, it is possible to demonstrate that the local log-minimum of an X-log convex and X-log quasi-convex function is also the global log-minimum. [ABSTRACT FROM AUTHOR]

We prove a weighted Lp boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space Lp(Rn ×Rm, ω1(x)dx, ω2(y)dy), provided that the weights ω1 and ω2 are certain radial weights and that the kernels are rough in the optimal space L(log L)(Sn-1 × Sm-1). In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted Lp boundedness of the related square and maximal functions. Our weighted Lp inequalities extend as well as generalize previously known Lp boundedness results. [ABSTRACT FROM AUTHOR]

This paper considers some preliminary conclusions of Fejér's integral inequality relevant to strongly geometric arithmetic convex functions that is a type of the class of convex functions and also a mapping to produce a novel trapezoidal form. This mapping is used to derive new theorems and results. By utilization these, some applications were given. [ABSTRACT FROM AUTHOR]

The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions. We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz constant. In addition, we also obtain the reverse of the superadditivity relation for a convex function, expressed in the same way. As an application, we obtain reverses of power mean inequalities, the Hölder inequality, and the Hermite-Hadamard inequality, expressed in terms of the Lipschitzianity. In particular, we derive reverses of the arithmetic-geometric mean inequality in both difference and quotient forms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies the single-machine group scheduling problem with convex resource allocation and learning effect. The number of jobs in each group is different, and the corresponding common due dates are also different, where the processing time of jobs follows a convex function of resource allocation. Under common due date assignment, the objective is to minimize the weighted sum of earliness, tardiness, common due date, resource consumption, and makespan. To solve the problem, we present the heuristic, simulated annealing, and branch-and-bound algorithms. Computational experiments indicate that the proposed algorithms are effective. [ABSTRACT FROM AUTHOR]

In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. We establish these theorems by creating duality models, which establish a relationship between SIVOPVC and its corresponding dual models, assuming generalized convexity conditions. Some examples are also given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, two mathematical methods are used for solving a complex multicriteria optimization problem as the considered convex differentiable vector optimization problem with vanishing constraints. First of them is the linearized approach in which, for the original vector optimization problem with vanishing constraints, its associated multiobjective programming problem is constructed at the given feasible solution. Since the aforesaid multiobjective programming problem constructed in the linearized method is linear, one of the existing methods for solving linear vector optimization problems is applied for solving it. Thus, the procedure for solving the considered differentiable vector optimization problems with vanishing constraints is presented in the paper. [ABSTRACT FROM AUTHOR]

This paper discussed the decay of a thermoelastic laminated beam subjected to nonlinear delay and nonlinear structural damping. We provided explicit and general energy decay rates of the solution by imposing suitable conditions on both weight delay and wave speeds. To achieve this, we leveraged the properties of convex functions and employed the multiplier technique as a specific approach to demonstrate our stability results. [ABSTRACT FROM AUTHOR]

Outer approximation (OA) for solving convex mixed-integer nonlinear programming (MINLP) problems is heavily dependent on the convexity of functions and a natural issue is to relax the convexity assumption. This paper is devoted to OA for dealing with a pseudo-convex MINLP problem. By solving a sequence of nonlinear subproblems, we use Lagrange multiplier rules via Clarke subdifferentials of subproblems to introduce a parameter and then equivalently reformulate such MINLP as the mixed-integer linear program (MILP) master problem. Then, an OA algorithm is constructed to find the optimal solution to the MNILP by solving a sequence of MILP relaxations. The OA algorithm is proved to terminate after a finite number of steps. Numerical examples are illustrated to test the constructed OA algorithm. [ABSTRACT FROM AUTHOR]

Environmental responsibility and economic benefits have promoted the development of closed-loop supply chains (CLSCs), and shortages and channels are considered to be two important issues in a CLSC. This paper explores the ordering and channel decisions in a CLSC with new and differentiated remanufactured products; considers the price and sales-effort-dependent demands, as well as the proportion of emergency orders determined by emergency order costs and backorder losses; and establishes integrated and decentralized CLSC game models. We introduce a stochastic sales effort, which affects two types of products. The numerical results show that sales effort and the order quantity of new and remanufactured products exhibit concave and convex functions, respectively. The upper limit of sales effort has a greater impact on supply chain decisions. High sales efforts can serve as a means of coordinating dispersed supply chains. Moreover, in different cases, the decisions of an integrated channel are better than those of a decentralized channel. Finally, whether the supply chain adopts an emergency order strategy depends on the relative cost of emergency orders and out-of-stock costs. According to this research, some management insights are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a three-step iterative scheme, called the MF-iteration process to approximate a common fixed point for a hybrid pair {τ, T} of single-valued and multi-valued maps satisfying a generalized contractive condition defined on uniformly convex Banach spaces. We establish the strong convergence theorem for the proposed process under some basic boundary conditions. We give a numerical example to prove our results’ convergence rate. Further, we compare the convergence speed of Sokhuma and Kaewkhao [29] and MF-iterations. we show numerically that the considered iterative scheme converges faster than Sokhuma and Kaewkhao [29] for single-valued and multi-valued non-expansive mappings. Our newly proven results generalize several relevant results in the literature. [ABSTRACT FROM AUTHOR]

This paper proves an equality for the case of twice-differentiable convex functions involving conformable fractional integrals. Using the established equality, we give new Simpson-type inequalities for the case of twice-differentiable convex functions via conformable fractional integrals. We also consider some special cases which can be deduced from the main results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the theoretical and numerical analysis of the two-step method, constructed as a modification of Polyak's heavy ball method with the inclusion of an additional momentum parameter. For the quadratic case, the convergence conditions are obtained with the use of the first Lyapunov method. For the non-quadratic case, sufficiently smooth strongly convex functions are obtained, and these conditions guarantee local convergence.An approach to finding optimal parameter values based on the solution of a constrained optimization problem is proposed. The effect of an additional parameter on the convergence rate is analyzed. With the use of an ordinary differential equation, equivalent to the method, the damping effect of this parameter on the oscillations, which is typical for the non-monotonic convergence of the heavy ball method, is demonstrated. In different numerical examples for non-quadratic convex and non-convex test functions and machine learning problems (regularized smoothed elastic net regression, logistic regression, and recurrent neural network training), the positive influence of an additional parameter value on the convergence process is demonstrated. [ABSTRACT FROM AUTHOR]

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures. [ABSTRACT FROM AUTHOR]

In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to the square root of the norm of the current gradient. For the basic scheme, as applied to the composite convex optimization problem, we establish the global convergence rate of the order O (k - 2) both in terms of the functional residual and in the norm of subgradients. Our main assumption on the smooth part of the objective is Lipschitz continuity of its Hessian. For uniformly convex functions of degree three, we justify global linear rate, and for strongly convex function we prove the local superlinear rate of convergence. Our approach can be seen as a relaxation of the Cubic Regularization of the Newton method (Nesterov and Polyak in Math Program 108(1):177–205, 2006) for convex minimization problems. This relaxation preserves the convergence properties and global complexities of the Cubic Newton in convex case, while the auxiliary subproblem at each iteration is simpler. We equip our method with adaptive search procedure for choosing the regularization parameter. We propose also an accelerated scheme with convergence rate O (k - 3) , where k is the iteration counter. [ABSTRACT FROM AUTHOR]

This paper delves into an inquiry that centers on the exploration of fractional adaptations of Milne-type inequalities by employing the framework of twice-differentiable convex mappings. Leveraging the fundamental tenets of convexity, Hölder's inequality, and the power-mean inequality, a series of novel inequalities are deduced. These newly acquired inequalities are fortified through insightful illustrative examples, bolstered by rigorous proofs. Furthermore, to lend visual validation, graphical representations are meticulously crafted for the showcased examples. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space. In this paper we show among others that, if the selfadjoint operators A and B satisfy the condition 0 < m ≤ A, B ≤ M, for some constants m, M, then... for all v ∈ [0;1]: We also have the inequalities for Hadamard product... for all v ∈ [0,1]. [ABSTRACT FROM AUTHOR]

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form. [ABSTRACT FROM AUTHOR]

This paper deals with a sharp version of mid-point type inequality in connection with fractional integrals of real valued absolutely continuous functions as a generalization and refinement of non-sharp classical mid-point inequality which is presented by the Riemann integrals of differentiable real valued functions whose derivative absolute values are convex. Some special functions, numerical means and a mid-point type formula are considered to discuss about some applications of main results. [ABSTRACT FROM AUTHOR]

In this paper the extension of the weighted Montgomery identity is established by using the integral formula of Pecaric, Matic and Ujevc. Further, by using this extended weighted Montgomery identity for functions whose derivatives of order n-1 are absolutely continunous functions, new inequalities of the weighted Hermite-Hadamard type are obtained. Also, applications of these results are given for various types of weight function. [ABSTRACT FROM AUTHOR]

This paper concerns the study singular integrals along twisted surfaces of the form We prove Lp bounds for the corresponding operators when the surfaces are defined by mappings more general than polynomials and convex functions, provided that the kernels are in L(logL)2(Sn-1x Sm-1) . [ABSTRACT FROM AUTHOR]

Integral inequalities are the proficient aspect of mathematical analysis. Various techniques have been deployed to acquire to fresh inequalities which are beneficial in various area problems. The aim of this paper is to derive some new analytic inequalities involving generalized weighted exponential beta functions. To attain our primary objectives, we introduce the generalized exponential function X and weighted form of exponential beta functions Furthermore, we briefly discuss their properties. we derive several inequalities in association with As the applications of these new developments,we conclude some error estimates of Ostrwoski's type inequalities, which show the significance of the obtained results. [ABSTRACT FROM AUTHOR]