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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]

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]

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]

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, 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 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 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 define the p -adic mixed Morrey type spaces and study the boundedness of p -adic fractional integral operators and their commutators on these spaces. More precisely, we first obtain the boundedness of p -adic fractional integral operators and their commutators on p -adic mixed central Morrey spaces. Moreover, we further extend these results on p -adic generalized mixed Morrey spaces, when a symbol function b belongs to the p -adic generalized mixed Campanato spaces. [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]

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]

In this paper, we explore a class of Hermite–Hadamard integral inequalities for convex and m-convex functions. The Hölder inequality is used to create this class, which has a wide range of applications in optimization theory. Some trapezoid-type inequalities and midpoint error estimates are investigated. Inequalities for several q-special functions are highlighted. As particular cases, we have included several previous results. [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]

This paper deals with generalizations and refinements of Fejér's inequality with some new applications. We introduce a real mapping M f ω (t) and obtain some its functional properties. We present several inequalities in connection with fractional integrals and their refinements by monotone functions. We also give some inequalities related to Euler's gamma and beta functions. Furthermore, we obtain some upper and lower bounds for generalized logarithmic mean and the expected value of random variables by using the arithmetic mean. [ABSTRACT FROM AUTHOR]

In this paper we deal with functions related to generalized convexity and refine Jensen type inequalities satisfied by such functions. Specifically, we extend inequalities satisfied by uniformly convex functions, strongly convex functions as well as superquadratic functions. [ABSTRACT FROM AUTHOR]

Inequalities play a major role in pure and applied mathematics. In particular, the inequality plays an important role in the study of Rosseland's integral for the stellar absorption. In this paper we obtain new Milne-type inequalities, and we apply them to the generalized Riemann–Liouville-type integral operators, which include most of the known Riemann–Liouville integral operators. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to present some new properties of convex functions, including Mercer-type, superadditive-type, and Jensen-type inequalities. Applications of these new properties will cover the celebrated Hermite–Hadamard inequality, Clarkson inequality, and the operator Mercer inequality. Comparisons with some existing results will be discussed too. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study certain inequalities for cr-log-h-convex functions with an interval value. To this end, we first give a definition of cr-log-h-convexity of interval-valued functions under the cr-order and study some properties of such functions. On this basis, we establish the Jensen-, Hermite–Hadamard-, and Fejér-type inequalities for cr-log-h-convex functions, and discuss some special cases. In addition, we give some numerical examples to illustrate the accuracy of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities involving both h 1 -convex functions and h 2 -concave functions such that (h 1 , h 2) is an increasing pair of functions. We apply the obtained results to Breckner's s-convex functions, P-convex functions and Godunova–Levin functions. In order to unify the results, we introduce generalized s-convex functions of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results. [ABSTRACT FROM AUTHOR]

In the present paper we prove a generalized version of the famous decomposition theorem of Ng. We also focus on the problem posed by Zsolt Páles concerning the Wright-convex functions. [ABSTRACT FROM AUTHOR]

The main goal of this paper is a study of Jensen-type inequalities based on Lidstone interpolation formula. We establish Jensen-type inequalities for convex functions of even order and for completely convex functions. Besides the interpolation formula, the main results are derived by using convexity properties of the Lidstone polynomials and the corresponding Green functions. In some cases, we obtain improved variants of the basic Jensen inequality. As an application, we derive some new power mean inequalities, as well as the corresponding Hölder-type inequality. Finally, we discuss Lah–Ribarič-type inequalities for the above classes of functions. Our results are then compared with the previously known from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Bullen-type generalized fractional integral identity for the differentiable functions and derive some new estimates for the Bullen-type fractional integral inequalities via the Raina fractional integrals. Further examples are given to indicate the validity of obtained results. Lastly, some error estimates for the quadrature rules are provided, and to find applications for our results to information theory, we proved a new generalization based on f-divergence measure. [ABSTRACT FROM AUTHOR]

In this paper, we conduct a research on a new version of the (p , q) -Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for (p , q) -midpoint and (p , q) -trapezoidal inequalities for special (p , q) -differentiable functions by using the notions of left and right (p , q) -derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some (p , q) -functions to confirm the correctness of newly established results. [ABSTRACT FROM AUTHOR]

In the paper, the authors investigate properties, including the monotonicity, logarithmic concavity, concavity, and inequalities, of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body. [ABSTRACT FROM AUTHOR]

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α -fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α -fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α -fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α -fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α -fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α -fractal functions. [ABSTRACT FROM AUTHOR]

In this paper we characterize random variables which are stable but not strictly stable in the sense of generalized convolution. We generalize the results obtained in Jarczyk and Misiewicz (J Theoret Probab 22:482-505, 2009), Misiewicz and Mazurkiewicz (J Theoret Probab 18:837-852, 2005), Oleszkiewicz (in Milman VD and Schechtman Lecture Notes in Math. 1807, Geometric Aspects of Functional Analysis (2003), Israel Seminar 2001–2002, Springer-Verlag, Berlin). The main problem was to find the solution of the following functional equation for symmetric generalized characteristic functions φ , ψ : ∀ a , b ≥ 0 ∃ c (a , b) ≥ 0 ∃ d (a , b) ≥ 0 ∀ t ≥ 0 φ (a t) φ (b t) = φ (c (a , b) t) ψ (d (a , b) t) , (A) where both functions c and d are continuous, symmetric, homogeneous but unknown. We give the solution of equation (A) assuming that for fixed ψ , c , d there exist at least two different solutions of (A). To solve (A) we also determine the functions that satisfy the equation (f (t (x + y)) - f (t x)) (f (x + y) - f (y)) = (f (t (x + y)) - f (t y)) (f (x + y) - f (x)) , (B) x , y , t > 0 , for a function f : (0 , ∞) → R . As an additional result we infer that each Lebesgue measurable or Baire measurable function f satisfying equation (B) is infinitely differentiable. [ABSTRACT FROM AUTHOR]

In this paper we prove new continuous refinements of some Jensen type inequalities in both direct and reversed forms. As applications we also derive some continuous refinements of Hermite–Hadamard, Hölder, and Popoviciu type inequalities. As particular cases we point out the corresponding results for sums and integrals showing that our results contain both several well-known but also some new results for these special cases. [ABSTRACT FROM AUTHOR]

In the current paper, a generalized Montgomery identity is obtained with the help of Taylor's formula on time scales. The obtained identity is used to establish Ostrowski inequality, mid-point inequality, and trapezoid inequality. Moreover, the weighted versions of generalized Montgomery identity and respective Ostrowski inequality are also discussed. Special cases are obtained for different time scales to obtain new and existing results. [ABSTRACT FROM AUTHOR]

By using superquadracity, the new inequalities in this paper generalize a result of Hardy–Littlewood–Pólya and refine Jensen's type inequalities. Also, inequalities related to rearrangements of sets are obtained. [ABSTRACT FROM AUTHOR]

Fractional calculus operators play a very important role in generalizing concepts of calculus used in diverse fields of science. In this paper, we use Riemann–Liouville fractional integrals to establish generalized identities, which are further applied to obtain midpoint and trapezoidal inequalities for convex function with respect to a strictly monotone function. These inequalities reproduce midpoint and trapezoidal inequalities for convex, harmonic convex, p-convex, and geometrically convex functions. Also, some new inequalities can be generated via specific strictly monotone functions. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Jensen-type inequality for generalized harmonically convex function on the fractal sets, and a generalized Jensen–Mercer inequality involving local fractional integrals is obtained. Moreover, we establish some generalized Jensen–Mercer-type local fractional integral inequalities for harmonically convex function. Also, we obtain some generalized related results using these inequalities on the fractal space. Finally, we give applications of generalized means and probability density function. [ABSTRACT FROM AUTHOR]

In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained. [ABSTRACT FROM AUTHOR]

In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher–Rao manifold of beta distributions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new Hermite–Hadamard-type inequalities for r-convex and geometrically convex functions and, additionally, some new Hermite–Hadamard-type inequalities by using the Hölder–İşcan integral inequality and an improved power-mean inequality. [ABSTRACT FROM AUTHOR]