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In this paper, we present some weighted Hölder-type inequalities for N −tuple of n × n complex matrices, with applications that include refined bounds for the norm of the sum of two matrices, the numerical radius of a matrix and the numerical radius for the product of two matrices. The relation with the existing literature will be discussed, where some results are proven to be improvements of celebrated results in the literature. Further, numerical examples are given to support the significance of other results. [ABSTRACT FROM AUTHOR]

In this research, a novel method for enhancing the Hölder–Işcan inequality through the utilization of both integrals and sums, as well as the mean power inequality, has been introduced. This approach outperforms traditional Hölder and mean power integral inequalities by employing a finite set of functions. Through the careful selection of the function ϕ$$ \phi $$, an entirely new category of classical inequalities emerges for both Hölder and mean power inequalities. [ABSTRACT FROM AUTHOR]

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting α = 0 = φ , γ = 1 , and w = 0 , σ (0) = 1 , λ = 1 , are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. [ABSTRACT FROM AUTHOR]

The focal point of this investigation is the exploration of solutions for Caputo-Hadamard fractional differential equations with boundary conditions, and it follows the initial formulation of a model that is intended to address practical problems. The research emphasizes resolving the challenges associated with determining precise solutions across diverse scenarios. The application of the Burton-Kirk fixed-point theorem and the Kolmogorov compactness criterion in Lp-spaces ensures the existence of the solution to our problem. Banach’s theory is crucial for the establishment of solution uniqueness, and it is complemented by utilizing the Hölder inequality in integral analysis. Stability ¨ analyses from the Ulam-Hyers perspective provide key insights into the system’s reliability. We have included practical examples, tables, and figures, thereby furnishing a comprehensive and multifaceted examination of the outcomes. [ABSTRACT FROM AUTHOR]

The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.

In this article, we have defined new weighted integral operators. We formulated a lemma in which we obtained a generalized identity through these integral operators. Using this identity, we obtain some new generalized Simpson's type inequalities for $(h,m)$-convex functions. These results we obtained using the convexity property, the classical Hölder inequality, and its other form, the power mean inequality. The generality of our results lies in two fundamental points: on the one hand, the integral operator used and, on the other, the notion of convexity. The first, because the ''weight'' allows us to encompass many known integral operators (including the classic Riemann and Riemann – Liouville), and the second, because, under an adequate selection of the parameters, our notion of convexity contains several known notions of convexity. This allows us to show that many of the results reported in the literature are particular cases of ours.

The focal point of this investigation is the exploration of solutions for Caputo-Hadamard fractional differential equations with boundary conditions, and it follows the initial formulation of a model that is intended to address practical problems. The research emphasizes resolving the challenges associated with determining precise solutions across diverse scenarios. The application of the Burton-Kirk fixed-point theorem and the Kolmogorov compactness criterion in $ {\mathfrak{L}}^\mathfrak{p} $-spaces ensures the existence of the solution to our problem. Banach's theory is crucial for the establishment of solution uniqueness, and it is complemented by utilizing the Hölder inequality in integral analysis. Stability analyses from the Ulam-Hyers perspective provide key insights into the system's reliability. We have included practical examples, tables, and figures, thereby furnishing a comprehensive and multifaceted examination of the outcomes.

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity. [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]

Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson's type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson's type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function. [ABSTRACT FROM AUTHOR]

The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions. We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz constant. In addition, we also obtain the reverse of the superadditivity relation for a convex function, expressed in the same way. As an application, we obtain reverses of power mean inequalities, the Hölder inequality, and the Hermite-Hadamard inequality, expressed in terms of the Lipschitzianity. In particular, we derive reverses of the arithmetic-geometric mean inequality in both difference and quotient forms. [ABSTRACT FROM AUTHOR]

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity.

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.

In this paper, we present a novel class of Copson-type inequalities involving the right-sided k-Hadamard fractional integral operator with all parameters of integrability p ≠ 0 and obtain some classical special cases of Copson inequalities. The main results will be proved by employing the Hölder inequality and the Fubini theorem.

The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.

In this article, general integral inequalities are obtained for functions whose absolute value of the second derivative is convex. These inequalities are more general versions of some results in the literature and we recaptured these results with the selection of special parameters. In the study, graphs are also used to compare the inequalities that occur with the change of the µ parameter. [ABSTRACT FROM AUTHOR]

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator Hölder type inequalities. Among other results, it is shown that if A = (At)t∈Ω and B=(Bt)t∈Ω are continuous fields of positive invertible operators in a unital C*-algebra A such that ∫Ω At dμ(t) ≤ IA and ∫Ω Bt dμ(t) ≤ IA, and if ωf is an arbitrary operator mean with the representing function f, then ...for all 0 < p ≤ 1, which is an extension of the operator Bellman inequality. [ABSTRACT FROM AUTHOR]

The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases m = 1 / 2 and m = 3 / 8 , as well as the two cases s = 1 and α = 1 , which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables. [ABSTRACT FROM AUTHOR]

Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities. [ABSTRACT FROM AUTHOR]

Different types of functions have been used in many various areas of mathematics until today, and today they are used to obtain new equality and inequalities in fractional analysis, which is one of the important areas of mathematics. In this study, we obtained integral inequalities for preinvex functions with the help of Atangana Baleanu fractional integral operator. [ABSTRACT FROM AUTHOR]

The main goal of this paper is a study of a class of more accurate Hilbert-type inequalities involving partial sums. We first derive a discrete Hilbert-type inequality involving two partial sums and the kernel 1/ max(mλ, nλ). Then, by virtue of the Hardy inequality, we establish a weaker version of the latter relation involving only one partial sum, as well as the corresponding equivalent form. In addition, we prove that the constants appearing on the right-hand sides of the established inequalities are the best possible. Finally, we also give integral analogues of the corresponding discrete results. [ABSTRACT FROM AUTHOR]

We formulate Hermite–Hadamard–Mercer type inequalities for ( A B K -FI) (Atangana–Baleanu Katagampula Fractional Integral) operators. We present new Dragomir–Agarwal type identities and establish a variants of Jensen–Hermite–Hadamard type inequalities involving A B K -FI operators for differentiable convex mappings. We present new bounds by employing Young and Hölder inequalities. We make links between our findings and a number of well-known discoveries in the literature. Finally, to support the effectiveness of our obtain results, we expose some inequalities examples in terms of Modified Bessel functions and matrices. The findings of this paper may lead to more research in this field. [ABSTRACT FROM AUTHOR]

The main motivation in this article is to prove new integral identities and related results. In this paper, we deal with E`-convex function, Hermite-Hadamard type inequalities, and Katugampola fractional integrals. Firstly, we established the Hermite-Hadamard type inequalities for E`-convex function via Katugampola fractional integrals operators. After that we obtain several new identities and give new results of Holder's inequality, Jensen's integral inequality, Young's inequality, and Power mean inequality for E`-convex function by using Katugampola fractional integral operators. We make connections of our results with various results recognized in the literature.

In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting α=0=φ,γ=1, and w=0,σ(0)=1,λ=1, are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field.

In this paper, we obtain necessary and sufficient conditions for the equivalence of two matrix summability methods. Some known results are also presented. [ABSTRACT FROM AUTHOR]

Convexity is a fundamental concept in analysis. Over the past few decades, many significant error bounds have been established for various quadrature rules using different types of convexity. This paper focuses on the Gauss–Radau quadrature formula. Initially, we introduce a novel identity related to 2-point left Radau-type rule. Next, we derive several integral inequalities for functions whose first derivatives are s-convex in the second sense. Finally, we present applications to special means to demonstrate the effectiveness of our results. [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 article, closed Newton–Cotes-type symmetrical inequalities involving four-point functions whose second derivatives are (s , m) -preinvex in the second sense are established. Some applications to quadrature formulas as well as inequalities involving special means are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the weighted variable exponent Lebesgue spaces defined on a probability space and give some information about the martingale theory of these spaces. We first prove several basic inequalities for expectation operators and obtain several norm convergence conditions for martingales in weighted variable exponent Lebesgue spaces. We discuss the Hölder inequality and embedding properties in these spaces. Finally, under some conditions we investigate Doob's maximal function. [ABSTRACT FROM AUTHOR]

Through this research, we introduce two new forms of the half-discrete Hilbert inequality for three variables. In addition, we show that the constants that appear on the right-hand of inequalities are the best. Also, we introduce the equivalence forms of the two inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a new parameterized identity. Based on this identity we establish some parametrized Simpson-like type symmetric inequalities, for functions whose first derivatives are s- t g s -convex via Reimann–Liouville frational operators. Some special cases are discussed. Applications to numerical quadrature are provided. [ABSTRACT FROM AUTHOR]

For all convex functions, the Hermite–Hadamard inequality is already well known in convex analysis. In this regard, Hermite–Hadamard and Ostrowski type inequalities were obtained using exponential type convex functions in this work. In addition, new generalizations were found for different values of θ. In conclusion, we believe that our work's technique will inspire more study in this field. [ABSTRACT FROM AUTHOR]

This paper demonstrates through a numerical experiment that utilization of strongly convex functions strengthens the bound presented for the Jensen gap in [4]. Consequently the improved result enables to present improvements in the bounds obtained for the Hölder and Hermite-Hadamard gaps and proposes such improvements in the results obtained for various entropies and divergences in information theory. [ABSTRACT FROM AUTHOR]

This paper investigates some well-known inequalities for q-h-integrals. These include Hölder, Ostrowski, Grüss and Opial type inequalities. Refinement of the Hadamard inequality for q-h-integrals is also established by applying the definition of strongly convex functions. From main theorems, q-Hölder, q-Ostrowski and q-Grüss inequalities can be obtained in particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we establish several Hölder-type inequalities using Jensen-type and Young-type inequalities as key tools. Particularly noteworthy is a reverse Hölder inequality with the Specht's ratio. Furthermore, we obtain a reverse Young-type inequality and we apply these results to the fractional context, both globally and locally. [ABSTRACT FROM AUTHOR]

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented.

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s=1, λ=α, σ(0)=1, and w=0. In conclusion, the methodology described in this article is expected to stimulate further research in this area.

Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson’s type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson’s type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function.

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly $ \left(s, m\right) $-convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means.

This article introduces extended (s,m)-prequasiinvex functions on coordinates, a new form of generalized convex function. Using a previously established identity, we derive new fractional Hermite-Hadamard type integral inequalities for functions whose mixed partial derivatives belong to this new class of functions. The obtained results generalize existing Hermite-Hadamard type inequalities and have numerous applications in mathematics and physics. To demonstrate the utility of our findings, we provide examples of applications to special means, such as arithmetic, harmonic and p-logarithmic means.

The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣f′∣| f^{\prime} | and ∣f′∣q(q>1)| f^{\prime} {| }^{q}\hspace{0.33em}\left(q\gt 1) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.

Abstract– In this paper, we have generalized two known theorems dealing with the absolute weighted arithmetic mean summability factors of infinite series and Fourier series to the absolute matrix summability methods. Some new and known results have also been obtained. [ABSTRACT FROM AUTHOR]

We demonstrated Hermite-Hadamard type results via r-convex function class. The proofs of results are based on the identities which are given in [16] and [3], established by Set et al. and Chun and Qi respectively. [ABSTRACT FROM AUTHOR]

The striking goal of this study is to introduce a q -identity for a parameterized integral operator via differentiable function. First, we discover the parameterized lemma for the q -integral. After that, we provide several q -differentiable inequalities. By taking suitable choices, some interesting results are obtained. With all of these, we displayed the findings from the traditional analysis utilizing q → 1 − . [ABSTRACT FROM AUTHOR]

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly -convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means. [ABSTRACT FROM AUTHOR]

In this paper, some open two-point Newton-Cotes type integral inequalities for functions whose first derivatives are convex via Riemann-Liouville fractional integrals are established. Our finding generalize some already known results. In order to illustrate the efficiency of our main results, some applications are given. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well. [ABSTRACT FROM AUTHOR]