Means Of Positive Real Numbers Research Articles

On some generalized Simpson type inequalities for (α,m)-coordinated convex functions in context of q 1 q 2-calculus

Abstract In the current investigation, we offer the generalized version of q 1 ⁢ q 2 {q_{1}q_{2}} -Simpson’s type inequalities via ( α , m ) {(\alpha,m)} -coordinated convex functions. To validate their generalized behavior, we demonstrate the link between our outcomes and the already derived ones. Moreover, we provide some application to special means of positive real numbers to support our findings. The principal outcomes raised in this investigation are extensions and generalizations of the comparable results in the history on Simpson’s inequalities for coordinated convex functions.

Properties of Coordinated h1,h2-Convex Functions of Two Variables Related to the Hermite–Hadamard–Fejér Type Inequalities

In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h1,h2-convex functions on the rectangle from the plane R2. Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some properties of two functionals which are connected with the coordinated h1,h2-convex functions are provided as well. Finally, we give applications of the acquired results to special means of positive real numbers.

On generalizations of post quantum midpoint and trapezoid type inequalities for (α,m)-convex functions

The aim of current study is to establish two crucial (p, q)b-integral identities for midpoint and trapezoid type inequalities. Utilizing these identities, we developed some new variant of midpoint and trapezoid type integral inequalities of differential (?,m)-convex functions using right post quantum integral approach. Moreover, we have presented the application of derived results related to special means of positive real numbers.

Some New Parameterized Quantum Fractional Integral Inequalities Involving s-Convex Functions and Applications

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behavior. Its significance is raised by the strong connection between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. By applying this identity, we obtain as main results some integral inequalities of trapezium, midpoint and Simpson’s type pertaining to s-convex functions. Moreover, we deduce several special cases, which are discussed in detail. To validate our theoretical findings, an example and application to special means of positive real numbers are presented. Numerical analysis investigation shows that the mixed fractional calculus with quantum calculus give better estimates compared with fractional calculus or quantum calculus separately.

Parameterized Quantum Fractional Integral Inequalities Defined by Using n-Polynomial Convex Functions

Convexity performs the appropriate role in the theoretical study of inequalities according to the nature and behaviour. There is a strong relation between symmetry and convexity. In this article, we consider a new parameterized quantum fractional integral identity. Following that, our main results are established, which consist of some integral inequalities of Ostrowski and midpoint type pertaining to n-polynomial convex functions. From our main results, we discuss in detail several special cases. Finally, an example and an application to special means of positive real numbers are presented to support our theoretical results.

Some General Fractional Integral Inequalities Involving LR–Bi-Convex Fuzzy Interval-Valued Functions

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

Refinement Mappings Related to Hermite-Hadamard Type Inequalities for GA-Convex Function

In this paper, we present some new refinement mappings associated with the Hermite–Hadamard type inequalities that are constructed for GA-convex mappings. Our investigation of the mappings leads to the discovery of several interesting features as well as the development of some inequalities for the Hermite–Hadamard type inequalities, which have already been established for GA-convex functions, as well as refining the relationship between the middle, rightmost, and leftmost elements of the function. Some applications to special means of positive real numbers are also given.

Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in q-Calculus

The main objective of this study is to establish two important right q-integral equalities involving a right-quantum derivative with parameter m∈[0,1]. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable (α,m)-convex functions. The fundamental benefit of these inequalities is that they may be transformed into q-midpoint- and q-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.

On inequalities of Hermite-Hadamard type via $ n $-polynomial exponential type $ s $-convex functions

<abstract><p>In this paper, a new class of Hermite-Hadamard type integral inequalities using a strong type of convexity, known as $ n $-polynomial exponential type $ s $-convex function, is studied. This class is established by utilizing the Hölder's inequality, which has several applications in optimization theory. Some existing results of the literature are obtained from newly explored consequences. Finally, some novel limits for specific means of positive real numbers are shown as applications.</p></abstract>

Simpson’s type inequalities for exponentially convex functions with applications

The Simpson's inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads of articles are bound for twice differentiable convex functions but nothing, to the best of our knowledge, is known yet for twice differentiable exponentially convex and quasi-convex functions. In this paper, we aim to do justice to this query. For this, we prove several Simpson's type inequalities for exponentially convex and exponentially quasi-convex functions. Our findings refine, generalize and complement existing results in the literature. We regain previously known results by taking \(\alpha=0\). In addition, we also show the importance of our results by applying them to some special means of positive real numbers and to Simpson's quadrature rule. The obtained results can be extended for different kinds of convex functions.

A Study of Uniform Harmonic χ -Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

In this paper, we introduce the notion of uniform harmonic χ -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.

Some New Post-Quantum Integral Inequalities Involving Twice (p, q)-Differentiable ψ-Preinvex Functions and Applications

The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p, q)-differentiable functions that are in absolute value bounded as well.

New Ostrowski-Type Fractional Integral Inequalities via Generalized Exponential-Type Convex Functions and Applications

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature.

New Inequalities of Fejer´ and Hermite-Hadamard type Concerning Convex and QuasiConvex Functions With Applications

This research contains new integral inequalities of Fejer and ´ Hermite-Hadamard type involving convex and quasi-convex functions. Applications of the newly established results for special means of positive real numbers are given.

Some Ostrowski type inequalities via $ n $-polynomial exponentially $ s $-convex functions and their applications

<abstract><p>This paper deals with introducing and investigating a new convex mapping namely, $ n $-polynomial exponentially $ s $-convex. Here, we present some algebraic properties and some logical examples to validate the theory of newly introduced convexity. Some novel adaptations of the well-known Hermite-Hadamard and Ostrowski type inequalities for this convex function have been established. Additionally, some special cases of the newly established results are derived as well. Finally, as applications some new limits for special means of positive real numbers are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature.</p></abstract>

Some new generalized $ \kappa $–fractional Hermite–Hadamard–Mercer type integral inequalities and their applications

<abstract><p>In this paper, we have established some new Hermite–Hadamard–Mercer type of inequalities by using $ {\kappa} $–Riemann–Liouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of Hermite–Hadamard–Mercer type via $ {\kappa} $–Riemann–Liouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.</p></abstract>

Matrix extremal problems and shift invariant means

We consider an m×m real symmetric matrix Ma(x) with a1,…,am on the main diagonal and x in all off-diagonal positions, where m≥2 and a=(a1,…,am) is a given m-tuple of positive real numbers. We study the extremal problem of finding the minimum and maximum of x where Ma(x) is positive semidefinite. We show that the polynomial det⁡Ma(x) in variable x has only real roots with a unique negative root and that Ma(x) is positive semidefinite (resp. definite) if and only if x lies in the closed (resp. open) interval determined by the negative and smallest positive roots. It is further shown that the negative and smallest positive root maps over m-tuples of positive real numbers contract the Thompson metric and induce new multivariate means of positive real numbers satisfying the monotonicity, homogeneity, joint concavity and super-multiplicativity. In particular, the smallest positive root map extends to such a mean of infinite variable of positive real numbers that realizes the limits for decreasing sequences, and it eventually gives rise to a shift invariant mean of bounded sequences.

Ostrowski type inequalities via the Katugampola fractional integrals

The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are $p-$convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.

Fejér type integral inequalities related with geometrically-arithmetically convex functions with applications

A new identity involving a geometrically symmetric function and a differentiable function is established. Some new Fejér type integral inequalities, connected with the left part of Hermite–Hadamard type inequalities for geometrically-arithmetically convex functions, are presented by using the Hölder integral inequality and the notion of geometrically-arithmetically convexity. Applications of our results to special means of positive real numbers are given.

NEW INEQUALITIES OF WIRTINGER TYPE FOR CONVEX AND MN-CONVEX FUNCTIONS

In this paper, we obtain some inequalities of Wirtinger type by using some classical inequalities and means for convex functions and establish some applications to special means for positive real numbers.