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This paper presents a novel approach by introducing a set of operators known as the left and right generalized tempered fractional integral operators. These operators are utilized to establish new Hermite–Hadamard inequalities for convex functions as well as the multiplication of two convex functions. Additionally, this paper gives two useful identities involving the generalized tempered fractional integral operator for differentiable functions. By leveraging these identities, our results consist of integral inequalities of the Hermite–Hadamard type, which are specifically designed to accommodate convex functions. Furthermore, this study encompasses the identification of several special cases and the recovery of specific known results through comprehensive research. Lastly, this paper offers a range of applications in areas such as matrices, modified Bessel functions and q -digamma functions. [ABSTRACT FROM AUTHOR]

The main results of this paper concern growth in sums of a k-convex function f. Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71–85, 2020)) of a growth result for f(A) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for | 2 k f (A) - (2 k - 1) f (A) |. We also generalise a recent result from Hanson et al. (J Lond Math Soc, 2021), proving that for any finite A ⊂ R | 2 k f (s A - s A) - (2 k - 1) f (s A - s A) | ≫ s | A | 2 s where s = k + 1 2 . This allows us to prove that, given any natural number s ∈ N , there exists m = m (s) such that if A ⊂ R , then 1 | (s A - s A) (m) | ≫ s | A | s. This is progress towards a conjecture (Balog et al. in Electron J Comb 24(3):Paper No. 3.14, 17, 2017) which states that (1) can be replaced with | (A - A) (m) | ≫ s | A | s. Developing methods of Solymosi, and Bloom and Jones, and using an idea from Bradshaw et al. (Electron J Comb 29, 2021), we present some new sum-product type results in the complex numbers C and in the function field F q ((t - 1)) . [ABSTRACT FROM AUTHOR]

In this paper, we study the functional that arises in numerous economic applications, in particular, in the monopolist problem. A special feature of these problems is that the domains of such functionals are nonclassical (in our case, increasing convex functions). We use an appropriate minimax theorem to prove the duality relation for . In particular, an important corollary is obtained stating that the dual functional (defined on a space of measures and known as the "Beckmann functional) attains its minimum. The present approach also provides simpler proofs of some previously known results. [ABSTRACT FROM AUTHOR]

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for q − h -integrals using properties of convex functions. Inequalities for q -integrals that have been published in recent years can be extracted from the main results of this paper. [ABSTRACT FROM AUTHOR]

Invesigation of bounds for Hankel determinat of analytic univalent functions is prominentintrest of many researcher from early twenth century to study geometric properties. Manyauthors obtained non sharp upper bound of third Hankel determinat for different subclasses ofanalytic univalent functions until Kwon et al. [5] obtained exact estimation of the fourth coefficeientof Carath'eodory class. Recently authors made use of an exact estimation of the fourthcoefficient, well known second and third coefficient of Carath'eodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions.Let w = f(z) = z + a2z² + · · · be analytic in the unit disk D = {z ∊ C: |z| < 1}, and Sbe the subclass of normalized univalent functions with f(0) = 0, and f'(0) = 1. Let z = f-1be the inverse function of f, given by f-1(w) = w + t2w² + · · · for some |w| < ro(f). LetSc ⊂ S be the subset of convex functions in D. In this paper, we estimate the best possibleupper bound for the third Hankel determinant for the inverse function z = f-1 when f ∊ Sc.Let Sc be the class of convex functions. We prove the following statements (Theorem): Iff ∊ Sc, then - H3,1(f-1)| = 1 36 and the inequality is attained for p0(z) = (1 + z³)/(1 - z³). [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions to the initial-boundary value problem for the following parabolic differential inclusion:$ \begin{equation*} \begin{cases} u_t\left(t, x \right) -\triangle _{p}u\left(t, x \right) \in -\partial \phi \left(u\left(t, x \right) \right) + G\left(t, x, u\left(t, x \right) \right) & (t, x) \in Q_T, \\ u(t, x) = 0 & (t, x) \in \Gamma_T, \\ u(0, x) = u_0(x) & x \in \Omega, \end{cases} \end{equation*} $where $ \Omega $ is a bounded open subset of $ \mathbb{R}^{N} $ with smooth boundary $ \partial \Omega, $ $ T>0 $, $ Q_{T}: = [0, T] \times \Omega $, $ \Gamma_T: = [0, T] \times \partial\Omega $, $ u_t = \frac{\partial u}{\partial t} $, $ \triangle_{p} $ is the $ p $-Laplace differential operator, $ \partial \phi $ denotes the subdifferential of a proper lower semicontinuous convex function $ \phi :\mathbb{R}\rightarrow \left[ 0, \infty \right] $, and $ G:Q_{T}\times \mathbb{R}\rightarrow 2^{ \mathbb{R}} \backslash \{\emptyset\} $ is a nonmonotone multivalued mapping.The case where $ \phi \equiv 0 $ and $ G(t, x, u) = |u|^{q-2}u $ gives the prototype of our problem, denoted by (E)$ _p $. The existence of time-local strong solutions for (E)$ _p $ is already studied by several authors. However, these results require a stronger assumption on $ q $ than that for the semi-linear case (E)$ _p $ with $ p = 2 $.More precisely, it has been long conjectured that (E)$ _p $ should admit a time-local strong solution for the Sobolev-subcritical range of $ q $, i.e., for all $ q \in (2, p^\ast) $ with $ p^\ast = \infty $ for $ p \geq N $ and $ p^\ast = \frac{N p}{N-p} $ for $ p

A real valued function f defined on a real open interval I is called Φ-convex if, for all x, y Є I, t Є [0,1] it satisfies f(tx+(1−t)y) ≤ tf(x)+(1−t)f(y)+tΦ((1−t)|x − y|)+(1−t)Φ(t|x − y|), where : R+→ R+ is a nonnegative error function. If f and -f are simultaneously Φ- convex, then f is said to be a Φ-affine function. In the main results of the paper, we describe the structural and inclusion properties of these two classes. We characterize these two classes of functions and investigate their relationship with approximately monotone and approximately- Hölder functions. We also introduce a subclass of error functions that enjoys the so-called à property and we show that the error function which is the most optimal for a Φ-convex function has to belong to this subclass. The properties of this subclass of error function are investigated as well. Then we offer two formulas for the lower Φ-convex envelop. Besides, a sandwich type theorem is also added. [ABSTRACT FROM AUTHOR]

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In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence. [ABSTRACT FROM AUTHOR]

By the asymptotic oracle property, non-convex penalties represented by minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) have attracted much attentions in high-dimensional data analysis, and have been widely used in signal processing, image restoration, matrix estimation, etc. However, in view of their non-convex and non-smooth characteristics, they are computationally challenging. Almost all existing algorithms converge locally, and the proper selection of initial values is crucial. Therefore, in actual operation, they often combine a warm-starting technique to meet the rigid requirement that the initial value must be sufficiently close to the optimal solution of the corresponding problem. In this paper, based on the DC (difference of convex functions) property of MCP and SCAD penalties, we aim to design a global two-stage algorithm for the high-dimensional least squares linear regression problems. A key idea for making the proposed algorithm to be efficient is to use the primal dual active set with continuation (PDASC) method to solve the corresponding sub-problems. Theoretically, we not only prove the global convergence of the proposed algorithm, but also verify that the generated iterative sequence converges to a d-stationary point. In terms of computational performance, the abundant research of simulation and real data show that the algorithm in this paper is superior to the latest SSN method and the classic coordinate descent (CD) algorithm for solving non-convex penalized high-dimensional linear regression problems. [ABSTRACT FROM AUTHOR]

It is undoubtedly that mathematical modelling and optimisation play a key role in the supply chain and the production management (SCPM). In this paper, we provide a survey on DC (Difference of Convex function) programming and DCA (DC Algorithm), a state-of-the-art optimisation approach for challenging problems in SCPM. DC programming and DCA constitute the backbone of non-convex programming and global optimisation. Whilst DC programming and DCA were widely and successfully investigated in many areas, it seems that they were not so much popular in the community of SCPM. There is therefore a need to further develop this efficient and scalable approach for SCPM applications, especially for large-scale problems in the context of Big data. For such purpose, this paper aims to present benchmark models and state-of-the-art DCA-based methods for solving challenging problems in SCPM systems. We prove that all the benchmark classes of optimisation models appeared in SCPM systems can be formulated/reformulated as a DC program and show how to solve these classes of problems by DCA-based algorithms. We offer the community of researchers in SCPM efficient algorithms in a unified DC programming framework to tackle various applications such as supply chain design, scheduling, multi-stage production/inventory system, vehicle routing, ... [ABSTRACT FROM AUTHOR]

Non-orthogonal multiple access (NOMA) cooperative communication technology can combine the advantages of NOMA and cooperative communication, providing high spectrum efficiency and increasing user coverage for next-generation wireless systems. However, the research on NOMA cooperative communication technology is still in a preliminary stage and has mainly concentrated on the scenario of fewer users. This paper focuses on a user-centered NOMA collaboration system in an ultra-dense network, and it constructs a resource allocation optimization problem to meet the demands of each user. Then, this paper decomposes the optimization problem into two subproblems; one is the grouping match among multiple relays and users, and the other is jointly allocating power and subcarrier resources. Accordingly, a dynamic packet matching algorithm based on Gale–Shapley and an iterative algorithm based on the difference of convex functions programing are proposed. Compared with existing schemes, the proposed algorithms can improve system throughput while ensuring the quality of service of users. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel distributed continuous-time algorithm aimed to solve optimization problems with cost function being a sum of local strictly convex multidimensional functions associated to individual agents. Additionally, the problems can have coupled equality and inequality constraints. We prove global asymptotic convergence of the algorithm for a connected graph topology. In order to investigate its practical implementation, we analyze convergence when Euler method is applied to represent discrete-time communication. Finally, we support our results with numerical experiments of the developed approach application for power balancing in New England power system. [ABSTRACT FROM AUTHOR]

In this paper, a new version of a higher-order strongly convex function is introduced which is named (ω-σ)-higherorder strongly convex function. Sufficient conditions for the existence of minimum for (ω - σ)-higher order strongly convex function is provided. The vector version of (ω-σ)-higher order strongly convex function is given and by using KKM theory an existence results for a solution of it is proved. Moreover, the compactness of the solution set of the vector version of (ω -σ)-higher order strongly convex function is investigated. The results of this article improve and extend the corresponding results presented in this area. [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]

Designing reasonable architectures of convolutional neural network (CNN) for specific image segmentation remains a challenging task, as the determination of the structure and hyperparameters of CNN depends heavily on expertise and requires a great deal of time. Evolutionary algorithm (EA) has been successfully applied to the automatic design of CNNs; however, the inherent stochastic search of EA tends to cause "experience loss" and requires very large computational resources. To deal with this problem, a maximal sparse convex surrogate model with updated empirical information is proposed in this paper to guide the evolutionary process of CNN design. This sparse convex function is transformed from a non-convex function to a maximized sparse convex function, which can better utilize the prior empirical knowledge to assist the evolutionary search. In addition, a balance strategy between computational resources and accuracy is proposed in the selection of reasonable network architectures. The proposed fully automatic design method of CNN is applied to the segmentation of steel microstructure images, and experimental results demonstrate that the proposed method is competitive with the existing state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we employ a q -Noor integral operator to perform a q -analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the q -fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the q -fractional integral operator and obtain some applications for the differential subordination. [ABSTRACT FROM AUTHOR]

Convexity is one of the fundamental principles of analysis. Over the past few decades, many important inequalities have been established for different classes of convex functions. In this paper, some Bullen-Simpson type integral inequalities for functions whose first derivatives are s-convex in the second sense are established. The cases where the first derivatives are bounded as well as Hölderian are also provided. Some applications to numerical integration and inequalities involving means are given. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite-Hadamard inequality for fractional integrals using harmonic convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Beta function. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value w0 = 0 on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels ki(i = 1, 2), namely k"i (t) = -ξi(t)Gi(-k'i(t)), where Gi are convex and increasing functions near the origin and ξi are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions ξi and Gi. These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature. [ABSTRACT FROM AUTHOR]

In large distributed systems, ensuring the efficient utilization of the available resources is a very challenging task. Given limited information regarding the state of the system and no centralized control over the outcome, decentralized scheduling mechanisms are unable to enforce optimal utilization. To better understand such systems, some classic papers that introduced game theoretic models used the "price of anarchy" measure to evaluate the system's performance. The paper "Resource-Aware Cost-Sharing Methods for Scheduling Games" by Christodoulou, Gkatzelis, and Sgouritsa overcomes some of the overly pessimistic results shown in this prior work by enhancing the scheduling mechanisms with access to some additional information regarding the state of the system: a "resource-aware" mechanism knows what machines are available in the system and uses this information to carefully incentivize the users toward more efficient Nash equilibrium outcomes. We study the performance of cost-sharing methods in a selfish scheduling setting where a group of users schedule their jobs on machines with load-dependent cost functions, aiming to minimize their own cost. Anticipating this user behavior, the system designer chooses a decentralized protocol that defines how the cost generated on each machine is to be shared among its users, and the performance of the protocol is evaluated over the Nash equilibria of the induced game. Previous work on selfish scheduling has focused on two extreme models: omniscient protocols that are aware of every machine and every job that is active at any given time, and oblivious protocols that are aware of nothing beyond the machine they control. We focus on a well-motivated middle-ground model of resource-aware protocols, which are aware of the set of machines in the system, but unaware of what jobs are active at any given time. Furthermore, we study the extent to which appropriately overcharging the users can lead to improved performance. We provide protocols that achieve small constant price of anarchy bounds when the cost functions are convex or concave, and we complement our positive results with impossibility results for general cost functions. Funding: This work was supported by the Royal Society [Grant LT140046], the Engineering and Physical Sciences Research Council [Grant EP/M008118/1], the National Science Foundation [Grants CCF-1161813, CCF-1216073, and CCF-1408635; CAREER Award CCF-2047907], and the Lise Meitner Award Fellowship. [ABSTRACT FROM AUTHOR]

In this paper, using a meromorphic univalent function of the form f(z) = 1 z−δ + ∞ k=j ak (z − δ)k, with missing coecients which has a simple pole at the point z = δ, 0 ≤ δ < 1, that is it is defined in U = U\ δ = z: z ∈ C and 0 < z − δ < 1, we note that for δ = 0 we obtain the class j of meromorphic functions. We define a new operator analogue of that of Aouf et al. [2] for univalent analytic functions defined in the open unit dick U and introduce new class of meromorphic univalent functions which for dierent values of its parameters many special new classes can be obtained from it. For this class of functions we obtained some of its properties such as coecient estimates, distortion theorem, the modified Hadamard product of two functions in it and also for function which its coecients are the sum of the squares of the coecients of two functions. Also we obtained analogue results of these results for each of the subclass obtained from this class. [ABSTRACT FROM AUTHOR]

In this article, we consider and investigate a split convex feasibility problem involving two bounded linear operators in Hilbert spaces. We introduce a self-adaptive CQ-type algorithm by selecting the stepsize which is independent of the operator norms and establish a strong convergence result of the proposed algorithm under some mild control conditions. Moreover, we propose a self-adaptive relaxed CQ-type algorithm for solving the problem constrained by sub-level sets of convex functions. A numerical example and an application in compressed sensing are also given to illustrate the convergence behaviour of our proposed algorithms. Our results in this paper improve and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]

The Mumford-Shah model is a classical segmentation model, but its objective function is nonconvex. The smoothing and thresholding (SaT) approach is a convex variant of the Mumford-Shah model, which seeks a smoothed approximation solution to the Mumford-Shah model. The SaT approach separates the segmentation into two stages: first, a convex energy function is minimized to obtain a smoothed image; then, a thresholding technique is applied to segment the smoothed image. The energy function consists of three weighted terms and the weights are called the regularization parameters. Selecting appropriate regularization parameters is crucial to achieving effective segmentation results. Traditionally, the regularization parameters are chosen by trial-and-error, which is a very time-consuming procedure and is not practical in real applications. In this paper, we apply a Bayesian inference approach to infer the regularization parameters and estimate the smoothed image. We analyze the convex variant Mumford-Shah variational model from a statistical perspective and then construct a hierarchical Bayesian model. A mean field variational family is used to approximate the posterior distribution. The variational density of the smoothed image is assumed to have a Gaussian density, and the hyperparameters are assumed to have Gamma variational densities. All the parameters in the Gaussian density and Gamma densities are iteratively updated. Experimental results show that the proposed approach is capable of generating high-quality segmentation results. Although the proposed approach contains an inference step to estimate the regularization parameters, it requires less CPU running time to obtain the smoothed image than previous methods. [ABSTRACT FROM AUTHOR]

In this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained. [ABSTRACT FROM AUTHOR]

In this paper, we find sufficient conditions to be imposed on the parameters of a class of functions related to the Barnes–Mittag-Leffler function that allow us to conclude that it possesses certain geometric properties (such as starlikeness, uniformly starlike (convex), strongly starlike (convex), convexity, and close-to-convexity) in the unit disk. The key tools in some of our proofs are the monotonicity properties of a certain class of functions related to the gamma function. [ABSTRACT FROM AUTHOR]

With the widespread application of spatial data in fields like econometrics and geographic information science, the methods to enhance the robustness of spatial econometric model estimation and variable selection have become a central focus of research. In the context of the spatial error model (SEM), this paper introduces a variable selection method based on exponential square loss and the adaptive lasso penalty. Due to the non-convex and non-differentiable nature of this proposed method, convex programming is not applicable for its solution. We develop a block coordinate descent algorithm, decompose the exponential square component into the difference of two convex functions, and utilize the CCCP algorithm in combination with parabolic interpolation for optimizing problem-solving. Numerical simulations demonstrate that neglecting the spatial effects of error terms can lead to reduced accuracy in selecting zero coefficients in SEM. The proposed method demonstrates robustness even when noise is present in the observed values and when the spatial weights matrix is inaccurate. Finally, we apply the model to the Boston housing dataset. [ABSTRACT FROM AUTHOR]

Let a normalized analytic function be given on the open unit disk. In this paper, we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes, we aim to find the upper bounds of the modules of the Hankel determinants obtained from the coefficients of the functions belonging to some classes defined by subordination. [ABSTRACT FROM AUTHOR]

The Hankel determinant H 2 , 1 F f - 1 / 2 of logarithmic coefficients is defined as H 2 , 1 F f - 1 / 2 : = Γ 1 Γ 2 Γ 2 Γ 3 = Γ 1 Γ 3 - Γ 2 2 , where Γ 1 , Γ 2 , and Γ 3 are the first, second, and third logarithmic coefficients of inverse functions belonging to the class S of normalized univalent functions. In this paper, we establish sharp inequalities H 2 , 1 F f - 1 / 2 ≤ 19 / 288 , H 2 , 1 F f - 1 / 2 ≤ 1 / 144 , and H 2 , 1 F f - 1 / 2 ≤ 1 / 36 for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and  (α , m) -convex, respectively. Two new q-integral identities are presented for three time q-differentiable functions. These lemmas are used like basic elements in our proofs, along with several important tools like q-power mean inequality, and q-Holder's inequality. In a special case, a non-trivial example is considered for a specific parameter and this case illustrates the investigated results. We make links between these findings and several previous discoveries from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of structured nonsmooth difference-of-convex (DC) minimization in which the first convex component is the sum of a smooth and a nonsmooth function, while the second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have weak convergence guarantees or need to solve lots of subproblems per iteration. Due to this, we propose hybrid algorithms for solving this problem in which we first compute approximate critical points and then check whether these points are approximate D-stationary points. Under suitable conditions, we prove that there exists a subsequence of iterates of which every accumulation point is a D-stationary point. Some preliminary numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc U = {z: z ∈ C and |z| < 1}. We find the bounds on the initial coefficients |c2| and |c3| and upper bounds for the Fekete-Szego functional for the functions in this class. [ABSTRACT FROM AUTHOR]

One of the fundamental inequalities which is used in many inequities is Jensen inequality. In fact, it is a base of some inequality such as the arithmetic mean, harmonic mean inequality also in inequality with respect to entropies and information theory. The purpose of this research paper is to give a new interesting refinement of Jensen inequality for two particular finite sequences by using uniformly convex function. Also, we give some applications of this in information theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider classes of approximate Minty and Stampacchia type vector variational inequalities using Clarke subdifferential on Hadamard manifolds and a class of nonsmooth multiobjective optimization problems. We investigate the relationship between the solution of these approximate vector variational inequalities and the solution of nonsmooth multiobjective optimization problems involving geodesic approximately convex functions. The results presented in this paper extend and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, by considering the identity established by Luo et al. in [C. Luo, T.-S. Du, M. Kunt and Y. Zhang, Certain new bounds considering the weighted Simpson-like type inequality and applications, J. Inequal. Appl. 2018 2018, Paper No. 332] and under the assumption of the quasi-convexity of the first derivative, we establish some new error estimates of the Simpson-like type inequalities. We also discuss the case where the first derivative satisfies the Hölder condition. At the end, we provide some applications to special means. The obtained results represent a continuation of the above-mentioned paper. [ABSTRACT FROM AUTHOR]

In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved. [ABSTRACT FROM AUTHOR]

The latest iterations of Simpson-type inequalities (STIs) are the topic of this paper. These inequalities were generated via convex functions and tempered fractional integral operators (TFIOs). To get these sorts of inequalities, we employ the well-known Hölder inequality and the inequality of exponent mean. The subsequent STIS are a generalization of several works on this topic that use the fractional integrals of Riemann-Liouville (FIsRL). Moreover, distinctive outcomes can be achieved through unique selections of the parameters. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the bounds of k-fractional integrals containing the Mittag-Leffter function. For establishing these bounds, a generalized convexity namely strongly exponentially (α, h-m)-p-convexity is utilized. The results of this article provide many new fractional inequalities for several types of fractional integrals and various kinds of convexities. Moreover, an identity is established which helps in proving a Hadamard type inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new refinement of Jensen-type inequality for uniformly convex functios. Furthermore, we apply those results in information theory and we obtain strong and more precise bounds for Shannon's entropy. [ABSTRACT FROM AUTHOR]

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ , m) -convex functions, s-convex functions, (s , r) -convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, M T -convex functions, P-convex functions, m-convex functions, (s , m) -convex functions, exponentially s-convex functions, (β , m) -convex functions, exponential-convex functions, ζ ¯ , β , γ , δ -convex functions, quasi-geometrically convex functions, s − e -convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the variation of functions has been defined, whose domain is a convex and compact set in the plane. Furthermore, in addition to presenting properties that satisfy this variation, the vector space formed by functions with finite variation is studied, demonstrating that it is a Banach space and its elements can be expressed as the difference of non-decreasing functions. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the Kantorovich optimal transportation problem with nonlinear cost functional generated by a cost function depending on the conditional measures of the transport plan. The case of a cost function nonconvex in the second argument is considered. It is proved that this nonlinear Kantorovich problem with general cost function on a Souslin space can be reduced to the same problem with a convex cost function. [ABSTRACT FROM AUTHOR]

Convex functions play a crucial role in various fields of mathematics, optimization, economics, and machine learning due to their distinctive properties and applications. In this paper, a new class of convex functions, called the RT-convex functions, is presented. Moreover, Hermite-Hadamard-type inequalities for the RT-convex functions are discussed. A number of applications of the RT-convex functions is also discussed. [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]

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. [ABSTRACT FROM AUTHOR]

This paper's major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided. [ABSTRACT FROM AUTHOR]

Let \mathcal {B} be the collection of rectangular parallelepipeds in \mathbb {R}^3 whose sides are parallel to the coordinate axes and such that \mathcal {B} consists of parallelepipeds with side lengths of the form s, 2^j s, t, where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right)\; \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*} On the other hand, if S is an infinite set, then the associated geometric maximal operator M_\mathcal {B} satisfies the weak type estimate \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right)^{2} \end{equation*} but does not satisfy an estimate of the form \begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right) \end{equation*} for any convex increasing function \phi : \mathbb [0, \infty) \rightarrow [0, \infty) satisfying the condition \begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*} [ABSTRACT FROM AUTHOR]

In this paper, we consider a multiple-sets split quasi-convex feasibility problem (MSSQFP), which is to find a point such that itself and its image under a linear transformation fall within two families of sublevel sets of quasi-convex functions in the space and the image space, respectively. A unified framework of the adaptive subgradient methods with general control schemes is proposed to solve the MSSQFP. This paper is contributed to establish the quantitative convergence theory of adaptive subgradient methods with several general control schemes. An interesting finding is disclosed by the iteration complexity results that the stochastic control enjoys both advantages of low computational cost requirement and low iteration complexity. In addition, a notion of the H¨older-type bounded error bound property for the MSSQFP is introduced, and the linear/sublinear convergence rates for the adaptive subgradient methods to a feasible solution of the MSSQFP is established. [ABSTRACT FROM AUTHOR]