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Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. [ABSTRACT FROM AUTHOR]

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]

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 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, several applications of the Hermite–Hadamard inequality for fractional integrals using G A -convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider different due-date assignment scheduling with group technology and resource allocation on a single machine, where the due date of each job may be different. Under constant processing times, the objective function is to minimize the scheduling cost (i.e., the weighted sum of earliness, tardiness, and due-date assignment cost, where the weights are position dependent). Under some optimal properties, we prove that this problem can be solved in  O (ζ log ζ)  time, where  ζ  is the number of jobs. The problem is also extended to cases which include linear and convex functions of the quantity of resource allocation. The objective function is minimizing the sum of the scheduling cost and the resource-consumption cost. For the special case of linear and convex functions, we show that the problem is polynomially solvable in  O (ζ 3)  time. [ABSTRACT FROM AUTHOR]

The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized. [ABSTRACT FROM AUTHOR]

The concrete strength criterion is the basis of strength analysis and evaluation under a complex stress state. In this paper, a large number of multiaxial strength tests were carried out, and many mathematical expressions of strength criteria were proposed based on the geometric characteristics and the assumption of a convex function. However, the rationality of the assumption of a convex function limits the use of these strength criteria. In particular, misjudgment will occur near the failure curve surface. Therefore, this paper does not assume the shape function of the criterion in advance. By collecting experimental data and using a machine learning method, it proposes a method of hidden function of failure curve surface. Based on 777 groups of experimental data, the random forest (RF), the back propagation neural network (BP) and the radial basis neural network (RBF) models were used to analyze and verify the feasibility and effectiveness of the method. Subsequently, the results were compared with the Ottosen strength criterion, the Guo Wang strength criterion and the Drucker–Prager (DP) strength criterion. The results show that the consistency between the strength criterion model established by the machine learning algorithm (especially random forest) and the experimental data is higher than the convex function multiaxis strength criterion of the preset failure surface shape. Moreover, the physical significance is clearer, the deficiency of the convex function failure surface hypothesis is avoided and the established multiaxial strength criterion of concrete is more universal. [ABSTRACT FROM AUTHOR]

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a 2 > 0 and without imposing any restrictive growth assumption on the damping term f 1 , using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function ψ , namely ψ ′ (t) ≤ − μ (t) G (ψ (t)) , where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G, as well as the function F defined by f 0 , which characterizes the growth behavior of f 1 at the origin. [ABSTRACT FROM AUTHOR]

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform  (B K , ρ) − invex, strictly uniform  (B K , ρ) − invex, uniform  (B K , ρ) − pseudoinvex, strictly uniform  (B K , ρ) − pseudoinvex, uniform  (B K , ρ) − quasiinvex and weakly uniform  (B K , ρ) − quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes S T * (γ , β) and K T (γ , β) of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class Y σ (V , W) associated of the operator I α , c ε defined by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which appeared recently in the literature on the geometric function theory of complex analysis. We also prove some simple conditions for starlikeness, convexity, and the strong starlikeness of several one-parameter families of integral operators, including (for example) a certain μ -convex integral operator and the familiar Bernardi integral operator. [ABSTRACT FROM AUTHOR]

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. [ABSTRACT FROM AUTHOR]

This paper mainly studies the distributed optimization problems in a class of undirected networks. The objective function of the problem consists of a smooth convex function and a non-smooth convex function. Each agent in the network needs to optimize the sum of the two objective functions. For this kind of problem, based on the operator splitting method, this paper uses the proximal operator to deal with the non-smooth term and further designs a distributed algorithm that allows the use of uncoordinated step-sizes. At the same time, by introducing the random-block coordinate mechanism, this paper develops an asynchronous iterative version of the synchronous algorithm. Finally, the convergence of the algorithms is proven, and the effectiveness is verified through numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text. [ABSTRACT FROM AUTHOR]

We investigate the classical optimal investment problem of the Merton model in a discrete time with market friction due to loss of wealth in trading. We consider the case of a finite number of investors, with the friction for each investor represented by a convex penalty function. This model cover the transaction costs and liquidity models studied previously in the literature. We suppose that each investor maximizes their utility function over all controls that keep the value of the portfolio after liquidation non-negative. In the main results of this paper, we prove the existence of an optimal strategy of investment by using a new approach based on the formulation of an equivalent general equilibrium economy model via constructing a truncated economy, and the optimal strategy is obtained using a classical argument of limits. [ABSTRACT FROM AUTHOR]

This paper deals with the strong approximate subdifferential formula for the difference of two vector convex mappings in terms of the star difference. This formula is obtained via a scalarization process by using the approximate subdifferential of the difference of two real convex functions established by Martinez-Legaz and Seeger, and the concept of regular subdifferentiability. This formula allows us to establish approximate optimality conditions characterizing the approximate strong efficient solution for a general DC problem and for a multiobjective fractional programming problem. [ABSTRACT FROM AUTHOR]

The field of position tracking control and communication engineering has been increasingly interested in time-varying quadratic minimization (TVQM). While traditional zeroing neural network (ZNN) models have been effective in solving TVQM problems, they have limitations in adapting their convergence rate to the commonly used convex activation function. To address this issue, we propose an adaptive non-convex activation zeroing neural network (AZNNNA) model in this paper. Using the Lyapunov theory, we theoretically analyze the global convergence and noise-immune characteristics of the proposed AZNNNA model under both noise-free and noise-perturbed scenarios. We also provide computer simulations to illustrate the effectiveness and superiority of the proposed model. Compared to existing ZNN models, our proposed AZNNNA model outperforms them in terms of efficiency, accuracy, and robustness. This has been demonstrated in the simulation experiment of this article. [ABSTRACT FROM AUTHOR]

In this paper, some estimates of third and fourth inequalities in Hermite–Hadamard-type inequalities for coordinated convex functions are proved using the non-additivity of the integrals and Fubini's theorem for fuzzy integrals. That is, the results are obtained in the fuzzy context and using the Lebesgue measure. Several examples are provided on how to evaluate these estimates in order to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new Simpson's type inequalities for coordinated convex functions by using post-quantum calculus. The results raised in this paper provide significant extensions and generalizations of other related results given in earlier works. [ABSTRACT FROM AUTHOR]

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions, (α , m) -convex functions, (α , m) -geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically (θ , m) -convex functions, m-harmonic harmonically convex functions, (s , r) -convex functions, arithmetic–geometric convex functions, logarithmically convex functions, (α , m) -logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable ϕ -convex functions, M T -convex functions, (s , m) -convex functions, p-convex functions, h-convex functions, σ -convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions, σ , s -convex functions, modified (p , h) -convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions, (α , h − m) − p -convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral, (k , s) -R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we propose two novel inertial forward–backward splitting methods for solving the constrained convex minimization of the sum of two convex functions, φ 1 + φ 2 , in Hilbert spaces and analyze their convergence behavior under some conditions. For the first method (iFBS), we use the forward–backward operator. The step size of this method depends on a constant of the Lipschitz continuity of ∇ φ 1 , hence a weak convergence result of the proposed method is established under some conditions based on a fixed point method. With the second method (iFBS-L), we modify the step size of the first method, which is independent of the Lipschitz constant of ∇ φ 1 by using a line search technique introduced by Cruz and Nghia. As an application of these methods, we compare the efficiency of the proposed methods with the inertial three-operator splitting (iTOS) method by using them to solve the constrained image inpainting problem with nuclear norm regularization. Moreover, we apply our methods to solve image restoration problems by using the least absolute shrinkage and selection operator (LASSO) model, and the results are compared with those of the forward–backward splitting method with line search (FBS-L) and the fast iterative shrinkage-thresholding method (FISTA). [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of infinite product of strongly quasi-nonexpansive mappings on geodesic spaces with curvature bounded above by one. Our main applications behind this study are to solve convex feasibility by alternating projections, and to solve minimizers of convex functions and common minimizers of several objective functions. To prove our main results, we introduce a new concept of orbital Δ -demiclosed mappings which covers finite products of strongly quasi-nonexpansive, Δ -demiclosed mappings, and hence is applicable to the convergence of infinite products. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator J μ , p . [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h 1 , h 2 -convex functions on the rectangle from the plane R 2 . Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some properties of two functionals which are connected with the coordinated h 1 , h 2 -convex functions are provided as well. Finally, we give applications of the acquired results to special means of positive real numbers. [ABSTRACT FROM AUTHOR]

In this paper, we present some new and novel mappings defined over 0 , 1 with the help of G A -convex functions. As a consequence, we obtain companions of Fejér-type inequalities for G A -convex functions with the help of these mappings, which provide refinements of some known results. The properties of these mappings are discussed as well. [ABSTRACT FROM AUTHOR]

Many researchers have been attracted to the study of convex analysis theory due to both facts, theoretical significance, and the applications in optimization, economics, and other fields, which has led to numerous improvements and extensions of the subject over the years. An essential part of the theory of mathematical inequalities is the convex function and its extensions. In the recent past, the study of Jensen–Mercer inequality and Hermite–Hadamard–Mercer type inequalities has remained a topic of interest in mathematical inequalities. In this paper, we study several inequalities for GA-h-convex functions and its subclasses, including GA-convex functions, GA-s-convex functions, GA-Q-convex functions, and GA-P-convex functions. We prove the Jensen–Mercer inequality for GA-h-convex functions and give weighted Hermite–Hadamard inequalities by applying the newly established Jensen–Mercer inequality. We also establish inequalities of Hermite–Hadamard–Mercer type. Thus, we give new insights and variants of Jensen–Mercer and related inequalities for GA-h-convex functions. Furthermore, we apply our main results along with Hadamard fractional integrals to prove weighted Hermite–Hadamard–Mercer inequalities for GA-h-convex functions and its subclasses. As special cases of the proven results, we capture several well-known results from the relevant literature. [ABSTRACT FROM AUTHOR]

In this paper, we found the error bounds for one of the open Newton–Cotes formulas, namely Milne's formula for differentiable convex functions in the framework of fractional and classical calculus. We also give some mathematical examples to show that the newly established bounds are valid for Milne's formula. [ABSTRACT FROM AUTHOR]

In this paper, a relaxed variable metric primal-dual fixed-point algorithm is proposed for solving the convex optimization problem involving the sum of two convex functions where one is differentiable with the Lipschitz continuous gradient while the other is composed of a linear operator. Based on the preconditioned forward–backward splitting algorithm, the convergence of the proposed algorithm is proved. At the same time, we show that some existing algorithms are special cases of the proposed algorithm. Furthermore, the ergodic convergence and linear convergence rates of the proposed algorithm are established under relaxed parameters. Numerical experiments on the image deblurring problems demonstrate that the proposed algorithm outperforms some existing algorithms in terms of the number of iterations. [ABSTRACT FROM AUTHOR]

Heuristic and evolutionary algorithms are proposed to solve challenging real-world optimization problems. In the evolutionary community, many benchmark problems for empirical evaluations of algorithms have been proposed. One of the most important classes of test problems is the class of convex functions, particularly the d-dimensional sphere function. However, the convex function type is somewhat limited. In principle, one can select a set of convex basis functions and use operations that preserve convexity to generate a family of convex functions. This method will inevitably introduce bias in favor of the basis functions. In this paper, the problem is solved by employing insights from computational geometry, which gives the first-ever general-purpose multi-dimensional convex landscape generator. The new proposed generator has the advantage of being generic, which means that it has no bias toward a specific analytical function. A set of N random d-dimensional points is generated for the construction of a d-dimensional convex hull. The upper part of the convex hull is removed by considering the normal of the polygons. The remaining part defines a convex function. It is shown that the complexity of constructing the function is O (M d 3) , where M is the number of polygons of the convex function. For the method to work as a benchmark function, queries of an arbitrary (d − 1) dimensional input are generated, and the generator has to return the value of the convex function. The complexity of answering the query is O (M d) . The convexity of the function from the generator is verified with a nonconvex ratio test. The performance of the generator is also evaluated using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) gradient descent algorithm. The source code of the generator is available. [ABSTRACT FROM AUTHOR]

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls under the umbrella of convexity. However, it is a significant theory that affects practically all areas of mathematics. In this paper, we introduce the notions of g , h -convexity or convexity with respect to a pair of functions on co-ordinates and discuss its fundamental properties. Moreover, we establish some novel Hermite–Hadamard- and Ostrowski-type inequalities for newly introduced co-ordinated convexity. Additionally, it is presented that the newly introduced notion of the convexity and given inequalities are generalizations of existing studies in the literature. Lastly, we look at various mathematical examples and graphs to confirm the validity of the newly found inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we aim to find sufficient conditions for the close-to-convexity of q-Bessel–Wright functions with respect to starlike functions, such as z 1 − z , z 1 − z 2 , and − log (1 − z) are in the open unit disc. Some consequences related to our main results are also included. [ABSTRACT FROM AUTHOR]

In this paper, a new deterministic method is proposed. This method depends on presenting (suggesting) some modifications to existing parameters of some conjugate gradient methods. The parameters of our suggested method contain a mix of deterministic and stochastic parameters. The proposed method is added to a line search algorithm to make it a globally convergent method. The convergence analysis of the method is established. The gradient vector is estimated by a finite difference approximation approach, and a new step-size h of this approach is generated randomly. In addition, a set of stochastic parameter formulas is constructed from which some solutions are generated randomly for an unconstrained problem. This stochastic technique is hybridized with the new deterministic method to obtain a new hybrid algorithm that finds an approximate solution for the global minimization problem. The performance of the suggested hybrid algorithm is tested in two sets of benchmark optimization test problems containing convex and non-convex functions. Comprehensive comparisons versus four other hybrid algorithms are listed in this study. The performance profiles are utilized to evaluate and compare the performance of the five hybrid algorithms. The numerical results show that our proposed hybrid algorithm is promising and competitive for finding the global optimum point. The comparison results between the performance of our suggested hybrid algorithm and the other four hybrid algorithms indicate that the proposed algorithm is competitive with, and in all cases superior to, the four algorithms in terms of the efficiency, reliability, and effectiveness for finding the global minimizers of non-convex functions. [ABSTRACT FROM AUTHOR]

With the development of science and technology, more and more data have been produced. For many of these datasets, only some of the data have labels. In order to make full use of the information in these data, it is necessary to classify them. In this paper, we propose a strong sparse quadratic kernel-free least squares semi-supervised support vector machine ( S S Q L S S 3 V M ), in which we add a ℓ 0 norm regularization term to make it sparse. An NP-hard problem arises since the proposed model contains the ℓ 0 norm and another nonconvex term. One important method for solving the nonconvex problem is the DC (difference of convex function) programming. Therefore, we first approximate the ℓ 0 norm by a polyhedral DC function. Moreover, due to the existence of the nonsmooth terms, we use the sGS-ADMM to solve the subproblem. Finally, empirical numerical experiments show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new versions of Hermite–Hadamard type inequalities for co-ordinated convex functions via q 1 , q 2 -integrals. Since the inequalities are newly proved, we therefore consider some examples of co-ordinated convex functions and show their validity for particular choices of q 1 , q 2 ∈ (0 , 1) . We hope that the readers show their interest in these results. [ABSTRACT FROM AUTHOR]

This paper aims to provide sufficient conditions for starlikeness and convexity of Hadamard product (convolution) of certain multivalent analytic functions with positive real parts. Moreover, the starlikeness conditions for a certain integral operator and other convolution results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new weighted Hermite–Hadamard-type inequalities for (n + 2) − convex functions by utilizing generalizations of Steffensen's inequality via Taylor's formula. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new line search technique, then employ it to construct a novel accelerated forward–backward algorithm for solving convex minimization problems of the form of the summation of two convex functions in which one of these functions is smooth in a real Hilbert space. We establish a weak convergence to a solution of the proposed algorithm without the Lipschitz assumption on the gradient of the objective function. Furthermore, we analyze its performance by applying the proposed algorithm to solving classification problems on various data sets and compare with other line search algorithms. Based on the experiments, the proposed algorithm performs better than other line search algorithms. [ABSTRACT FROM AUTHOR]

As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind. [ABSTRACT FROM AUTHOR]

In this paper, a new operator D s f , with s a real number, is defined considering functions that belong to the known class of p-valent analytic functions in the open unit disk U. Applying this operator, a new subclass of p-valently analytic functions is introduced and some interesting subordination- and coefficient-related properties of the functions in this class are discussed. It is also shown that for certain values given to the parameters involved in the definition of the class, p-valently starlike and p-valently convex functions of certain orders can be obtained, respectively. Examples are also given as applications of the newly proven results. [ABSTRACT FROM AUTHOR]

The entropy of the partition generated by an n-tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition and its complementary, here referred to as the bias, allow us to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales with limited bias because their tones are distributed along specific interval fractions of a regular partition. The most typical case in music concerns partitions associated with well-formed scales generated by a single tone h. These scales are maximal even sets that combine two elementary intervals. Then, the normalized entropy depends on each number of intervals as well as their relative size. When well-formed scales are refined, several nested families stand out with increasing regularity. It is proven that a scale of minimal bias, i.e., with less bias than those with fewer tones, is always a best rational approximation of l o g 2 h . [ABSTRACT FROM AUTHOR]

In this paper, we present some new refinements of Hermite–Hadamard inequalities for continuous convex functions by using (p , q) -calculus. Moreover, we study some new (p , q) -Hermite–Hadamard inequalities for multiple integrals. Many results given in this paper provide extensions of others given in previous research. [ABSTRACT FROM AUTHOR]

In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by φ -cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results. [ABSTRACT FROM AUTHOR]

In the paper, we prove Steffensen type inequalities for positive finite measures by using functions which are convex in point. Further, we prove Steffensen type inequalities on Borel σ -algebra for the function of the form f / h which is convex in point. We conclude the paper by showing that these results also hold for convex functions. [ABSTRACT FROM AUTHOR]

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval [ a 0 , a 1 ] × [ c 0 , c 1 ] ⊆ ℜ 2 , we introduce the notion of partial q θ -, q ϕ -, and q θ q ϕ -symmetric derivatives and a q θ q ϕ -symmetric integral. Moreover, we will construct the q θ q ϕ -symmetric Hölder's inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications. [ABSTRACT FROM AUTHOR]

The notion of strong differential subordination was introduced in 1994 and the theory related to it was developed in 2009. The dual notion of strong differential superordination was also introduced in 2009. In a paper published in 2012, the notion of strong differential subordination was given a new approach by defining new classes of analytic functions on U × U ¯ having as coefficients holomorphic functions in U ¯ . Using those new classes, extended Sălăgean and Ruscheweyh operators were introduced and a new extended operator was defined as L α m : A n ζ * → A n ζ * , L α m f (z , ζ) = (1 − α) R m f (z , ζ) + α S m f (z , ζ) , z ∈ U , ζ ∈ U ¯ , where R m f (z , ζ) is the extended Ruscheweyh derivative, S m f (z , ζ) is the extended Sălăgean operator and A n ζ * = { f ∈ H (U × U ¯) , f (z , ζ) = z + a n + 1 ζ z n + 1 + ⋯ , z ∈ U , ζ ∈ U ¯ }. This operator was previously studied using the new approach on strong differential subordinations. In the present paper, the operator is studied by applying means of strong differential superordination theory using the same new classes of analytic functions on U × U ¯. Several strong differential superordinations concerning the operator L α m are established and the best subordinant is given for each strong differential superordination. [ABSTRACT FROM AUTHOR]

Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal of this article is to establish a novel and generalized identity for the Caputo–Fabrizio fractional operator. With the help of this specific developed identity, we derive new fractional integral inequalities via exponential convex functions. Furthermore, we give an application to some special means. [ABSTRACT FROM AUTHOR]

In this study, we introduce the new subclasses, M α (sin) and M α (cos) , of α -convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class M α (cos) for the case n = 3 , proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied. [ABSTRACT FROM AUTHOR]