Special Means Of Real Numbers Research Articles

Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

Harmonic conformable refinements of Hermite-Hadamard Mercer inequalities by support line and related applications

ABSTRACT We establish new conformable fractional Hermite-Hadamard (H–H) Mercer type inequalities for harmonically convex functions using the concept of support line. We introduce two new conformable fractional auxiliary equalities in the Mercer sense and apply them to differentiable functions with harmonic convexity. We also use Power-mean, Hölder’s and improved Hölder inequality to derive new Mercer type inequalities via conformable fractional integrals. The accuracy and superiority of the offered technique are clearly depicted through impactful visual illustrations. We also use our technique to derive new estimates for hypergeometric functions and special means of real numbers that are more precise than existing ones. Some applications are provided as well. Our results generalize and extend some existing ones in the literature.

Parametrized multiplicative integral inequalities

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.

On the multiparameterized fractional multiplicative integral inequalities

AbstractWe introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus.

On parameterized inequalities for fractional multiplicative integrals

Abstract In this article, we present a one-parameter fractional multiplicative integral identity and use it to derive a set of inequalities for multiplicatively s s -convex mappings. These inequalities include new discoveries and improvements upon some well-known results. Finally, we provide an illustrative example with graphical representations, along with some applications to special means of real numbers within the domain of multiplicative calculus.

Analysis and Applications of Some New Fractional Integral Inequalities

This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis.

Earthquake convexity and some new related inequalities

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

Hermite–Hadamard type inequalities for multiplicative Riemann–Liouville fractional integrals

In this paper, we present a multiplicative fractional integral identity. Based upon it, we establish the Hermite–Hadamard type inequalities for multiplicatively convex functions via multiplicative Riemann–Liouville fractional integrals. We also give some applications to special means of real numbers in the framework of multiplicative calculus. To make the results obtained here more intuitive for the readers, we provide some examples for suitable choices of multiplicative convex functions and their graphical descriptions.

Some Further Results Using Green’s Function for s -Convexity

For s-convex functions, the Hermite–Hadamard inequality is already well-known in convex analysis. In this regard, this work presents new inequalities associated with the left-hand side of the Hermite–Hadamard inequality for s-convexity by utilizing a novel technique based on Green’s function. Also, Hölder, Young, and power-mean inequalities are used to obtain these new inequalities. Finally, some applications to special means of real numbers are provided. In conclusion, we think that the methodology used in this work will encourage more research in this field.

Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus.

A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

Study of quantum Ostrowski's-type inequalities for differentiable convex functions

UDC 517.9 We prove some new q-Ostrowski's-type inequalities for differentiable and bounded functions. Moreover, we present the relationship between the newly established and already known inequalities, which is very interesting for new readers. Some applications to special means of real numbers are given to make the results more valuable.

On certain Ostrowski type integral inequalities for convex function via AB-fractional integral operator

<abstract><p>We investigate and prove a new lemma for twice differentiable functions with the fractional integral operator $ AB $. Based on this newly developed lemma, we derive some new results about this identity. These new findings provide some generalizations of previous findings. This research builds on a novel new auxiliary result that allows us to create new variants of Ostrowski type inequalities for twice differentiable convex mappings. Some of the newly presented results' special cases are also discussed. As applications, several estimates involving special means of real numbers and Bessel functions are depicted.</p></abstract>

HERMITE–HADAMARD TYPE INEQUALITIES FOR KATUGAMPOLA FRACTIONAL INTEGRALS

In the paper, basing on the Katugampola fractional integrals ${}^\rho\mathcal{K}^\alpha_{a+}f$ and ${}^\rho\mathcal{K}^\alpha_{b-}f$ with $f\in\mathfrak{X}_c^p(a,b)$, the authors establish the Hermite-Hadamard type inequalities for convex functions, give their left estimates, and apply these newly-established inequalities to special means of real numbers. When $\rho\to1$, these results become the corresponding ones for the Riemann-Liouville fractional integrals.

Semi P-geometric-arithmetically functions and some new related inequalities

In this manuscript, the authors introduce the concept of the semi P-geometric-arithmetically functions (semi P-GA functions) and give their some algebraic properties. Then, they get Hermite-Hadamard?s integral inequalities for semi P-GA-functions (geometric-arithmetically convex). In addition, the authors obtain new inequalities by using H?lder and H?lder-??can integral inequalities with the help of an identity. Then, the aouthors compare the results obtained with both H?lder, H?lder-??can integral inequalities and prove that the H?lder-??can integral inequality gives a better approximation than the H?lder integral inequality. Also, some applications to special means of real numbers are also given.

SOME BULLEN-TYPE INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS

In this paper, we establish some new Bullen-type inequalities for differentiable convex functions using the generalized fractional integrals. The main advantage of the inequalities and operators used to obtain them is that these inequalities can be turned into some existing inequalities for Riemann integrals and new inequalities for Riemann–Liouville fractional integral inequalities and [Formula: see text]-fractional integrals. Finally, we add some applications of special means of real numbers using the newly established inequalities to make these results more interesting.

On n-fractional polynomial P-functions

In this paper, we introduce and study the concept of n-fractional polynomial P-functions and establish Hermite-Hadamard?s inequalities for this type of functions. In addition, we obtain some new Hermite-Hadamard type inequalities for functions whose first derivative in absolute value is n-fractional polynomial P-functions by using H?lder and power-mean integral inequalities. We also extend our initial results to functions of several variables. Next, we point out some applications of our results to give estimates for the approximation error of the integral the function in the trapezoidal formula and for some inequalities related to special means of real numbers.

Some new midpoint and trapezoidal type inequalities in multiplicative calculus with applications

In this paper, we use multiplicative twice differentiable functions and establish two new multiplicative integral identities. Then, we use convexity for multiplicative twice differentiable functions and establish some new midpoint and trapezoidal type inequalities in the framework of multiplicative calculus. Finally, we give some applications to special means of real numbers to make these inequalities more interesting for the readers.

NEW INEQUALITIES OF HERMITE–HADAMARD TYPE FOR n-POLYNOMIAL s-TYPE CONVEX STOCHASTIC PROCESSES

The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the [Formula: see text]-polynomial [Formula: see text]-type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter [Formula: see text]. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the [Formula: see text]-polynomial [Formula: see text]-type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering.

Certain fractional inequalities via the Caputo Fabrizio operator

The Caputo Fabrizio fractional integral operator is one of the key concepts in fractional calculus. It is involved in many concrete and practical issues. In the present study, we have discussed some novel ideas to fractional Hermite-Hadamard inequalities within a Caputo Fabrizio fractional integral framework. The fractional integral under investigation is used to establish some new fractional Hermite-Hadamard inequalities. The findings of this study can be seen as a generalization and extension of numerous earlier inequalities via convex function. In addition, we demonstrate a few applications of our findings to special means of real numbers.