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In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper "Monotonicity properties for a ratio of finite many gamma functions" published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided. [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]

This paper is concerned with the probability distribution a n , j : = 4 - n 2 j j 2 n - 2 j n - j , j = 0 , 1 , ⋯ , n . We present basic properties of the sequence (a n , j) : integral representations, recurrence formulas, convexity properties, bounds for the associated information potential. Two related random variables are also studied and compared from the point of view of the stochastic convex ordering and in connection with strongly convex functions. We consider a quadrature formula and investigate it with analytic and probabilistic methods. The paper contains two open problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers' comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus. [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]

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σ m of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients | a m + 1 | and | a 2 m + 1 | are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications. [ABSTRACT FROM AUTHOR]

In its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce four new generalized quantum integrals of two variables functions which are called T q -integrals. We also give some examples of each of these given integrals. Then we establish corresponding T q -Hermite–Hadamard type inequalities for co-ordinated convex functions. In order to show correctness of our results, we apply our main theorems to several co-ordinated convex functions. [ABSTRACT FROM AUTHOR]

The authors of the paper suggest a novel approach in order to examine an integral equality using conformable fractional operators. By using this identity, some Newton-type inequalities are proved for differentiable convex functions by taking the modulus of the newly established equality. Moreover, we prove some Newton-type inequalities by using the Hölder and power-mean inequality. Furthermore, some new results are presented by using special choices of obtained inequalities. Finally, we give some conformable fractional Newton-type inequalities for functions of bounded variation. [ABSTRACT FROM AUTHOR]

In this paper we introduce necessary and sufficient conditions for a real-valued function to be preinvex. Some properties of preinvex functions and new versions of Jensen's integral type inequality in this setting are given. Several examples are also involved. [ABSTRACT FROM AUTHOR]

The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended (G ′ G 2) -expansion method, two reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions used the definition of the truncated M-derivative (TMD) to understand its dynamical behaviour. The hyperbolic, periodic, and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, and singular solitary wave solitons are obtained. We observe the fractional parameter impact of the above derivative on the physical phenomena. Each set of travelling wave solutions have a symmetrical mathematical form. Last but not least, we employ Mathematica to produce 2D and 3D figures of the analytical soliton solutions to emphasize the influence of TMD on the behaviour and symmetry of the solutions for the proposed problem. The physical importance of the solutions found for particular values of the combination of parameters during the representation of graphs as well as understanding of physical incidents. [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 develop a concept of a logarithmically superquadratic function. Such a class of functions is defined via superquadratic functions. We first establish some new properties of superquadratic functions. In particular, we derive the corresponding superadditivity relation and its reverse, as well as the external form of the Jensen inequality and its reverse. Then, as a direct consequence of the established results, we obtain the corresponding properties for logarithmically superquadratic functions. Further, we show that logarithmically superquadratic functions with values greater than or equal to one are convex and logarithmically superadditive. In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions. [ABSTRACT FROM AUTHOR]

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field. [ABSTRACT FROM AUTHOR]

In this paper, we establish several new inequalities for q-differentiable coordinated convex functions that are related to the right side of Hermite–Hadamard inequalities for coordinated convex functions. We also show that the inequalities proved in this paper generalize the results given in earlier works. Moreover, we give some examples in order to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the Δ 2 , which will be useful to obtain the convexity results. We examine the correlation between the positivity of ( w 0 RL Δ α f) (t) and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of (2 , 3) , H k , ϵ and M k , ϵ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of ( w 0 RL Δ α f) (t) and convexity of the function on a finite time set N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , ... , P } for some P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , ... } . The numerical part of the paper is dedicated to examinin the validity of the sets H k , ϵ and M k , ϵ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new (p , q) -integral inequalities of Simpson's second type for preinvex functions. Many results given in this paper provide generalizations and extensions of the results given in previous research. Moreover, some examples are given to illustrate the investigated results. [ABSTRACT FROM AUTHOR]

This paper adopts different optimization algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization Algorithm (PSO-Algorithm) to train Back-Propagation (BP) neural networks, fits the Chinese, the Czech, Slovak, Hungarian, and Polish gross domestic product (GDP) growth model (from 1995 to 2020) and makes short-term simulation predictions. We use the PSO-Algorithm and GA with strong global search ability to optimize the weights and thresholds of the network, combine them with the BP neural network, and apply the resulting Particle Swarm Optimization Back-Propagation (PSO-BP) combined model or Genetic-Algorithm Back-Propagation (GA-BP) combined model to allow the network to achieve fast convergence. Besides, we also compare the above two hybrid models with standard multivariate regression model and BP neural network with different initialization methods like normal uniform and Xavier for fitting and short-term simulation predictions. Finally, we obtain the excellent results that all the above models have achieved a good fitting effect and PSO-BP combined model on the whole has a smaller error than others in predicting GDP values. Through the technology of PSO-BP and GA-BP, we have a clearer understanding of the five countries gross domestic product growth trends, which is conducive to the government to make reasonable decisions on the economic development. [ABSTRACT FROM AUTHOR]

This paper aims to extend the application of the Ostrowski inequality, a crucial tool for figuring out the error bounds of various numerical quadrature rules, including Simpson's, trapezoidal, and midpoint rules. Specifically, we develop a more comprehensive class of Ostrowski-type inequalities by utilizing the weighted version of Riemann–Liouville (RL) fractional integrals on an increasing function. We apply our findings to estimate the error bounds of Hadamard-type inequalities. Our results are more comprehensive, since we obtain the results of the existing literatures as particular cases for certain parameter values. This research motivates researchers to apply this concept to other fractional operators. [ABSTRACT FROM AUTHOR]

This paper proves an equality for the case of differentiable convex functions including the conformable fractional integrals. By using this equality, we establish several parameterized inequalities with the help of the conformable fractional integrals. Several inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly present some completely monotonic properties and new inequalities involving the k-gamma and the k-polygamma functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the concept of multiplicatively p-convex functions and established Hermite-Hadamard type integral inequalities in the setting of multiplicative calculus for this newly created class of functions. We also gave integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively p-convex functions. Furthermore, we obtained novel multiplicative integral-based inequalities for the product and quotient of convex and multiplicatively p-convex functions. Additionally, we derived certain upper limits for this new class of functions. The findings we proved are generalizations of the results in the literature. The results obtained in this study may inspire further research in various scientific areas. [ABSTRACT FROM AUTHOR]

In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of preinvex functions which is called general h-harmonic preinvex for real-valued stochastic processes. Further, we define the multidimensional general h-harmonic preinvex stochastic processes. We prove the Hermite-Hadamard inequality and obtain some important results for these processes. Some previous results are special cases of the results obtained in this paper. [ABSTRACT FROM AUTHOR]

Unfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it "earthquake convexity". In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Hölder, Hölder–İşcan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Hölder–İşcan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of strong invexity with respect to the E-operator. We discuss the necessary and sufficient conditions for the considered E-minimax fractional programming problem. Wolfe dual has been presented for the considered problem, and weak, strong, and strict converse duality theorems have been proved. Also, an application of the results derived is discussed in the context of a E-multiobjective fractional programming problem. At suitable places, nontrivial examples have been formulated with appropriate justification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new Hermite-Hadamard type inequality by using post-quantum calculus for s-preinvex function and establish a new (p,q)-integral identity to derive some new Ostrowski type inequality for (p,q)-differentiable function. We give some application to special means for our newly proved inequalities. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-s-convex fuzzy-interval-valued functions (log-s-convex fuzzy-IVFs). By this concept, we have introduced Hermite–Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Moreover, some new Hermite–Hadamard–Fejér inequalities (HH–Fejér-inequalities) and Jensen's inequalities via log-s-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area. [ABSTRACT FROM AUTHOR]

The class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including fundamental inequalities, supermultiplicative type inequalities, Jensen-Mercer inequality, integral inequalities, and refined forms. The obtained results extend some celebrated results from the context of convexity to geometric convexity, with interesting applications to numerical inequalities for the hyperbolic and exponential functions. [ABSTRACT FROM AUTHOR]

In this paper, a new integral identity is provided. Based on this equality, Simpson-type dual integral inequalities for functions whose first derivatives are s-convex via Riemann–Liouville fractional integrals are established. [ABSTRACT FROM AUTHOR]

In the last two decades, several properties of operators that are weaker than monotonicity have received attention by researchers from many areas including mathematical economics, with the goal to develop new tools applicable in convex analysis and related topics. This paper puts in perspective notions that are extensions of monotoniticity but not beyond quasimonotonicity like pseudomonotonicity, semistrict quasimonotonicity, strict quasimonotonicity and proper quasimonotonicity, and discusses systematically when the sum of two operators satisfying one of those properties, inherits the same property. The case of properly quasimonotone operators deserves a special attention since this notion, being stronger than quasimonotonicity, suffices to obtain many results, including the solvability of variational inequality problems. Several examples showing the optimality in some sense of our results, are presented. [ABSTRACT FROM AUTHOR]

We study the connections between the equation ∑ i = 1 n a i f (α i x + (1 - α i) y) = 0 and the corresponding inequality. ∑ i = 1 n a i f (α i x + (1 - α i) y) ≥ 0. At present, it is clear that linear functional equations should be solved with the use of the results due to L. Székelyhidi. We show that the simplest and most efficient way of dealing with the (continuous) solutions of linear inequalities of the above form is connected with the use of stochastic orderings tools. It will be shown that, in order to solve the inequality, we need to know the continuous solutions of the corresponding equation. In the last part of the paper, we obtain some simple but unexpected connections in the other direction. [ABSTRACT FROM AUTHOR]

We consider sequences of Steklov type operators and an associated functional equation. For a suitable sequence, we establish asymptotic formulas. [ABSTRACT FROM AUTHOR]

We present an adaptive method of approximate integration of convex (as well as concave) functions based on a certain refinement of the celebrated Hermite–Hadamard inequality. Numerical experiments are performed and the role of harmonic numbers is shown. [ABSTRACT FROM AUTHOR]

In this paper, we look at the general form of the two-point quadrature formulas, which doesn't have to be symmetric. Under the convexity constraint of the first derivative, we propose a new scheme that gives the best approximation of the two-point quadrature rules. [ABSTRACT FROM AUTHOR]

In this paper, based on a given parameterized identity that generates a quadrature rule family similar to Simpson's second formula, we establish some new Simpson-like type inequalities for functions with bounded as well as Lipchitzian derivatives from which we can deduce the famous 3/8-Simpson's inequality. The study concludes with an application example from management science. [ABSTRACT FROM AUTHOR]

In this paper, new inequalities for the left and right sides of the Hermite–Hadamard inequality are acquired for twice-differentiable mappings. Conformable fractional integrals are used to derive these inequalities. Furthermore, we provide our results by using special cases of obtained theorems. [ABSTRACT FROM AUTHOR]