Trapezoid Type Inequalities Research Articles

Certain exponential type $ m $-convexity inequalities for fractional integrals with exponential kernels

<abstract><p>By applying exponential type $ m $-convexity, the Hölder inequality and the power mean inequality, this paper is devoted to conclude explicit bounds for the fractional integrals with exponential kernels inequalities, such as right-side Hadamard type, midpoint type, trapezoid type and Dragomir-Agarwal type inequalities. The results of this study are obtained for mappings $ \omega $ where $ \omega $ and $ |\omega'| $ (or $ |\omega'|^q $with $ q\geq 1 $) are exponential type $ m $-convex. Also, the results presented in this article provide generalizations of those given in earlier works.</p></abstract>

On Hermite-Hadamard type inequalities associated with the generalized fractional integrals

In this paper, we obtain new generalization of Hermite-Hadamard inequalities via generalized fractional integrals defined by Sarikaya and Ertu?ral in [12]. We establish some midpoint and trapezoid type inequalities for functions whose first derivatives in absolute value are convex involving generalized fractional integrals.

Some trapezoid type inequalities for generalized fractional integral

In this paper, we have established some trapezoid type inequalities for generalized fractional integral. The results presented here would provide some fractional inequalities and Riemann-Liouville type fractional operators.

Hermite-Hadamard and Ostrowski type inequalities in $ \mathfrak{h} $-calculus with applications

<abstract><p>In this paper, we prove Hermite-Hadamard inequality for convex functions in the framework of $ \mathfrak{h} $-calculus. We also use the notions of $ \mathfrak{h} $-derivative and $ \mathfrak{h} $-integral to prove Ostrowski's and trapezoidal type inequalities for bounded functions. It is also shown that the newly established inequalities are the generalization of the comparable inequalities in the literature. Finally, using some examples, we demonstrate the validity of newly formed inequalities and show how they can be used to special means of real numbers.</p></abstract>

Some fractional integral inequalities via $ h $-Godunova-Levin preinvex function

<abstract><p>In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for $ h $-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries.</p></abstract>

Ostrowski type integral inequalities, weighted Ostrowski, and trapezoid type integral inequalities with powers

In this paper, we prove some new weighted Ostrowski and Trapezoid type inequalities with powers on time scales. The results will be proved by employing the generalized version of Montgomery identity with weights on time scales designed and proved for this purpose. As special cases, we will derive some new weighted discrete inequalities of Ostrowski and Trapezoid types which to the best of the authors' knowledge are essentially new.

A new extension of quantum Simpson's and quantum Newton's type inequalities for quantum differentiable convex functions

In this paper, we prove two identities involving quantum derivatives, quantum integrals, and certain parameters. Using the newly proved identities, we prove new inequalities of Simpson's and Newton's type for quantum differentiable convex functions under certain assumptions. Moreover, we discuss the special cases of our main results and obtain some new and existing Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities, and trapezoidal type inequalities.

On Some New Fractional Ostrowski- and Trapezoid-Type Inequalities for Functions of Bounded Variations with Two Variables

In this paper, we first prove three identities for functions of bounded variations. Then, by using these equalities, we obtain several trapezoid- and Ostrowski-type inequalities via generalized fractional integrals for functions of bounded variations with two variables. Moreover, we present some results for Riemann–Liouville fractional integrals by special choice of the main results. Finally, we investigate the connections between our results and those in earlier works. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

On Some New Trapezoidal Type Inequalities for Twice (p, q) Differentiable Convex Functions in Post-Quantum Calculus

Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and symmetry, has various applications for quantum calculus. Inequalities has a strong association with convex and symmetric convex functions. In this study, first we establish a p,q-integral identity involving the second p,q-derivative and then we used this result to prove some new trapezoidal type inequalities for twice p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of some existing results in the field of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

Some Parameterized Quantum Midpoint and Quantum Trapezoid Type Inequalities for Convex Functions with Applications.

Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and entropy, has various applications for quantum calculus. Inequalities and entropy functions have a strong association with convex functions. In this study, we prove quantum midpoint type inequalities, quantum trapezoidal type inequalities, and the quantum Simpson’s type inequality for differentiable convex functions using a new parameterized q-integral equality. The newly formed inequalities are also proven to be generalizations of previously existing inequities. Finally, using the newly established inequalities, we present some applications for quadrature formulas.

General Norm Inequalities of Trapezoid Type for Fréchet Differentiable Functions in Banach Spaces

In this paper we establish some error bounds in approximating the integral by general trapezoid type rules for Fréchet differentiable functions with values in Banach spaces.

New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions

The main objective of this paper is to introduce I p , q ϱ -derivative and I p , q ϱ -integral for interval-valued functions and discuss their key properties. Also, we prove the I p , q ϱ -Hermite–Hadamard inequalities for interval-valued functions is the development of p , q ϱ -Hermite–Hadamard inequalities by using new defined I p , q ϱ -integral. Moreover, we prove some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. It is also shown that the results presented in this paper are extensions of some of the results already shown in earlier works. The proposed studies produce variants that would be useful for performing in-depth investigations on fractal theory, optimization, and research problems in different applied fields, such as computer science, quantum mechanics, and quantum physics.

Some New Hermite-Hadamard and Related Inequalities for Convex Functions via (p,q)-Integral.

In this investigation, for convex functions, some new –Hermite–Hadamard-type inequalities using the notions of derivative and integral are obtained. Furthermore, for -differentiable convex functions, some new () estimates for midpoint and trapezoidal-type inequalities using the notions of integral are offered. It is also shown that the newly proved results for and can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities.

On generalized Ostrowski, Simpson and Trapezoidal type inequalities for co-ordinated convex functions via generalized fractional integrals

In this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we offer some generalized inequalities for differentiable co-ordinated convex functions with a rectangle in the plane mathbb{R} ^{2}. Furthermore, by special choice of parameters in our main results, we obtain several well-known inequalities such as the Ostrowski inequality, trapezoidal inequality, and the Simpson inequality for Riemann and Riemann–Liouville fractional integrals.

On parameterized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integrals

The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers.

Some Inequalities for a New Class of Convex Functions with Applications via Local Fractional Integral

The understanding of inequalities in convexity is crucial for studying local fractional calculus efficiency in many applied sciences. In the present work, we propose a new class of harmonically convex functions, namely, generalized harmonically ψ - s -convex functions based on fractal set technique for establishing inequalities of Hermite-Hadamard type and certain related variants with respect to the Raina’s function. With the aid of an auxiliary identity correlated with Raina’s function, by generalized Hölder inequality and generalized power mean, generalized midpoint type, Ostrowski type, and trapezoid type inequalities via local fractional integral for generalized harmonically ψ - s -convex functions are apprehended. The proposed technique provides the results by giving some special values for the parameters or imposing restrictive assumptions and is completely feasible for recapturing the existing results in the relative literature. To determine the computational efficiency of offered scheme, some numerical applications are discussed. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.

Sharp Estimation Type Inequalities for Lipschitzian Mappings in Euclidean Sense on a Disk

Some sharp trapezoid and midpoint type inequalities for Lipschitzian bifunctions defined on a closed disk in Euclidean sense are obtained by the use of polar coordinates. Also, bifunctions whose partial derivative is Lipschitzian are considered. A new presentation of Hermite-Hadamard inequality for convex function defined on a closed disk and its reverse are given. Furthermore, two mappings H t and h t are considered to give some generalized Hermite-Hadamard type inequalities in the case that considered functions are Lipschitzian in Euclidean sense on a disk.

Dynamical significance of generalized fractional integral inequalities via convexity

<abstract> The main goal of this paper is to develop the significance of generalized fractional integral inequalities via convex functions. We obtain the new version of fractional integral inequalities with the extended Wright generalized Bessel function acting as a kernel for the convex function, which deals with the Hermite-Hadamard type and trapezoid type inequalities. Moreover, we establish new mid-point type and trapezoid type integral inequalities for $ (\eta_{1}, \eta_{2}) $-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for $ (\eta_{1}, \eta_{2}) $-convex function related to Fejér type. The results discussed in this paper are a generalized version of many inequalities in literature. </abstract>

On generalizations of quantum Simpson's and quantum Newton's inequalities with some parameters

&lt;abstract&gt;&lt;p&gt;In this paper, we prove two identities concerning quantum derivatives, quantum integrals, and some parameters. Using the newly proved identities, we prove new Simpson's and Newton's type inequalities for quantum differentiable convex functions with two and three parameters, respectively. We also look at the special cases of our key findings and find some new and old Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities, and trapezoidal type inequalities.&lt;/p&gt;&lt;/abstract&gt;

Generalized integral inequalities for ABK-fractional integral operators

<abstract> In this paper, we employ new version of the Atangana-Baleanu integral operator namely $ ABK $-fractional integrals to obtain two general integral identities complying second-order derivatives for a given function. Thus allowing us to derive new generalized Hermite-Hadamard type inequalities via $ ABK $- fractional integrals. Moreover we give several new versions of Mid-point and Trapezoid type inequalities by employing Hölder, Young and Jensen inequalities. </abstract>