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In this paper, we prove some new discrete inequalities with weights of Hardy’s type and employ them to establish some properties of discrete weighted Gehring class of sequences satisfying reverse Hölder inequality. We also consider the weighted class of sequences satisfying a reverse inequality for nonincreasing sequences and prove some higher summability theorems. [ABSTRACT FROM AUTHOR]

In this research, we define an essential identity for differentiable functions in the framework of tempered fractional integral. By utilizing this identity, we deduce several modifications of fractional Milne-type inequalities. We provide novel expansions of Milne-type inequalities in the domain of tempered fractional integrals. The investigation emphasises important functional categories, including convex functions, bounded functions, Lipschitzian functions, and functions with bounded variation. [ABSTRACT FROM AUTHOR]

In this article, we study some Hermite-Hadamard-type inequalities for strongly hh-convex functions on co-ordinates in Rn{{\mathbb{R}}}^{n}, which extend some known results. Some mappings connected with these inequalities and related applications are also obtained.

In this research, we investigate the error bounds associated with Milne’s formula, a well-known open Newton–Cotes approach, initially focused on differentiable convex functions within the frameworks of fractional calculus. Building on this work, we investigate fractional Milne-type inequalities, focusing on their application to the more refined class of twice-differentiable convex functions. This study effectively presents an identity involving twice differentiable functions and Riemann–Liouville fractional integrals. Using this newly established identity, we established error bounds for Milne’s formula in fractional and classical calculus. This study emphasizes the significance of convexity principles and incorporates the use of the Hölder inequality in formulating novel inequalities. In addition, we present precise mathematical illustrations to showcase the accuracy of the recently established bounds for Milne’s formula. [ABSTRACT FROM AUTHOR]

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

In this study, some mappings related to the Fejér-type inequalities for GAGA-convex functions are defined over the interval [0,1]{[}0,1]. Some Fejér-type inequalities for GAGA-convex functions are proved using these mappings. Properties of these mappings are considered and consequently we obtain refinements of some known results.

In this paper, we give some new sharp bounds for sinc and hyperbolic sinc functions via cosine and hyperbolic cosine functions, which these bounds refine or improve most of recent published results. [ABSTRACT FROM AUTHOR]

This paper investigates a technique that uses Riemann-Liouville fractional integrals to study several Euler-Maclaurin-type inequalities for various function classes. Afterwards, we provide our results by using special cases of obtained theorems and This paper is to derive examples. Moreover, we give some Euler-Maclaurin-type inequalities for bounded functions by fractional integrals. Furthermore, we construct some fractional Euler-Maclaurin-type inequalities for Lipschitzian functions. Finally, we offer some Euler-Maclaurin-type inequalities by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

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

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

In the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section. [ABSTRACT FROM AUTHOR]

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

Inspired by the idea and technique in a recent paper of Huy et al. (Linear Algebra Appl 656: 368–384, 2023), Ighachane and Bouchangour showed an important refinement of the results by Sababheh (Linear Algebra Appl 506: 588–602, 2016) involving both convex and log-convex functions. In this paper, we continue to establish further new inequalities for both convex and log-convex functions with some applications to scalar and matrix means as well as matrix norm and numerical radius. The significance of the obtained results is two folded: the results themselves and the way they extend many known results in the literature. [ABSTRACT FROM AUTHOR]

In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs. [ABSTRACT FROM AUTHOR]

Using the differential operator Ω s : = s I - x d d x , s > 0 , we build a new class of infinitely divisible distributions on the half-line. For this class, we give a stochastic interpretation and we provide several monotonicity properties for the associated subordinators. As an application, we solve a problem raised separately by Sendov and Shan in (J Theor Probab 28:1689–1725, 2015) and by Simon in (Math Nachr 285(4): 497–506, 2012) on the distribution of the stable subordinators. Finally, we provide a new complete monotonicity property for the Mittag-Leffler function. [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]

In this current research, we focus on the domain of tempered fractional integrals, establishing a novel identity that serves as the cornerstone of our study. This identity paves the way for the Milne-type inequalities, which are explored through the framework of differentiable convex mappings inclusive of tempered fractional integrals. The significance of these mappings in the realm of fractional calculus is underscored by their ability to extend classical concepts into more complex, fractional dimensions. In addition, by using the Hölder inequality and power-mean inequality, we acquire some new Milne-type inequalities. Moreover, the practicality and theoretical relevance of our findings are further demonstrated through the application of specific cases derived from the theorems. [ABSTRACT FROM AUTHOR]

Fractional calculus is a crucial foundation in mathematics and applied sciences, serving as an extremely valuable tool. Besides, the new hybrid fractional operator, which combines proportional and Caputo operators, offers better applications in numerous fields of mathematics and computer sciences. Due to its wide range of applications, we focus on the proportional Caputo-hybrid operator in this research article. Firstly, we begin by establishing a novel identity for this operator. Then, based on the newfound identity, we establish some integral inequalities that are relevant to the left-hand side of Hermite–Hadamard-type inequalities for the proportional Caputo-hybrid operator. Furthermore, we show how the results improve upon and refine many previous findings in the setting of integral inequalities. Later, we present specific examples together with their related graphs to offer a better understanding of the newly obtained inequalities. Our results not only extend previous studies but also provide valuable viewpoints and methods for tackling a wide range of mathematical and scientific problems. [ABSTRACT FROM AUTHOR]

We give conditions in order to approximate a real function by $\phi$-strongly modulus $c$ convex functions. [ABSTRACT FROM AUTHOR]

In this study, we concentrate on solving the problem of non-Lipschitz absolute value equations (NAVE). A new Bezier curve based smoothing technique is introduced and a new Levenberg–Marquardt type algorithm is developed depending on the smoothing technique. The numerical performance of the algorithm is analysed by considering some well-known and randomly generated test problems. Finally, the comparison with other methods is illustrated to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Inspired by a lemma of Schmidt (Proc Am Math Soc 124:3003–3013, 1996) we present several inequalities involving max 1 ≤ j ≤ n | 1 - x j | , max 1 ≤ j ≤ n | log (x j) | and ∑ i = 1 n | 1 - x i | r 1 / r , ∑ i = 1 n | log (x i) | r 1 / r. [ABSTRACT FROM AUTHOR]

Generalizations of some Steffensen's inequalities for the class of higher order convex functions are obtained in present study. To prove the results, some preliminary lemmas are established by using extended Montgomery identities via Taylor's formula on time scales. Furthermore, Ostrowski type inequalities are also established, along with prove of optimization of the constant. Moreover, newly obtained results are discussed in discrete and quantum calculus as well.

In this paper, we determine the monotonicity of three classes of functions involving modified Bessel functions of the second kind. We also presented some inequalities and bounds for modified Bessel functions of the second kind. In particular, our results derive bounds for the Airy function and its derivative. [ABSTRACT FROM AUTHOR]

By using an integral arithmetic mean, a generalization of Levinson's inequality given in (Pečarić et al. in Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering, vol. 187, 1992) and results from (Vukelić in Appl. Anal. Discrete Math. 14:670–684, 2020), we give extension of Wulbert's result from (Wulbert in Math. Comput. Model. 37:1383–1391, 2003). Also, we obtain inequalities with divided differences for the functions of higher order. [ABSTRACT FROM AUTHOR]

Several bounds of trigonometric-exponential and hyperbolic-exponential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions.

The main idea of this article is the investigation of atmospheric internal waves, often known as gravity waves. This arises within the ocean rather than at the interface. A shallow fluid assumption is illustrated by a series of nonlinear partial differential equations in the framework. Because the waves are scattered over a wide geographical region, this system can precisely replicate atmospheric internal waves. In this research, the numerical solutions to the fuzzy fourth-order time-fractional Boussinesq equation (BSe) are determined for the case of the aquifer propagation of long waves having small amplitude on the surface of water from a channel. The novel scheme, namely the generalized integral transform (proposed by H. Jafari [35]) coupled with the Adomian decomposition method (GIADM), is used to extract the fuzzy fractional BSe in R,Rn and (2nth)-order including gH-differentiability. To have a clear understanding of the physical phenomena of the projected solutions, several algebraic aspects of the generalized integral transform in the fuzzy Caputo and Atangana-Baleanu fractional derivative operators are discussed. The confrontation between the findings by Caputo and ABC fractional derivatives under generalized Hukuhara differentiability are presented with appropriate values for the fractional order and uncertainty parameters ℘∈[0,1] were depicted in diagrams. According to proposed findings, hydraulic engineers, being analysts in drainage or in water management, might access adequate storage volume quantity with an uncertainty level.

In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral. [ABSTRACT FROM AUTHOR]

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

We show that certain known or new inequalities for the logarithm of circular hyperbolic functions imply bounds for exp(±x2) proved in [1].

In this study, we define for the first time the non-Lipschitz generalization of absolute value equations and concentrate on solving the problem of non-Lipschitz absolute value equations based on smoothing techniques. Two different types of smoothing techniques which are local and global ones are considered in smoothing process of the problem. With the help of these smoothing techniques, the non-Lipschitz absolute value equations are reformulated as a family of parametrized smooth equations. Two new algorithms are developed to solve the problem by the help of smoothing functions. Finally, the numerical experiments have been performed to illustrate the efficiency of the new algorithms. [ABSTRACT FROM AUTHOR]

For a rational function r(z) having no zeros outside a unit disk, Li et al. (J Lond Math Soc 51:523–531, 1995) proved: | r ′ (z) | ≥ | B ′ (z) | | r (z) | , for | z | = 1 , where B (z) is the Blashke product. This result was improved and generalized in various ways and a recent generalization was obtained by Mir (Indian J Pure Appl Math 51:749–760, 2020). In this paper, we prove some results which include as special cases many Turán-type polynomial inequalities proved earlier. [ABSTRACT FROM AUTHOR]

In this paper, we present certain class of strictly positive definite functions related to the extended Hurwitz-Lerch Zeta function, digamma function and the modified Bessel function of first kind. The key tools in our proofs are some explicit formulas for the modified Bessel function of the first kind and a new integral representation of the extended Hurwitz-Lerch Zeta function when their terms contain the Fox H-function. As applications, monotonicity properties and some new functional inequalities of the aforementioned functions are established. [ABSTRACT FROM AUTHOR]

Let ψ be the digamma function and let L(a,b) = (b − a)/log(b/a) be the logarithmic mean of a and b. We prove that the inequality holds for all real numbers a and b with b > a ≥ α0. Here, α0 ≈ 0.56155 is the only positive solution of The constant lower bound α0 is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a ≥ 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0, [ABSTRACT FROM AUTHOR]

In this article, an identity is proved for the case of twice-differentiable functions whose second derivatives in absolute value are convex. With the help of this equality, several corrected Euler–Maclaurin-type inequalities are established using the Riemann–Liouville fractional integrals. Several important fractional inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, the results are presented using special cases of obtained theories. [ABSTRACT FROM AUTHOR]