Derivatives In Absolute Value Research Articles

Generalization of Some Integral Inequalities for Arithmetic Harmonically Convex Functions

In this study, by using an integral identity, Hölder integral inequality and modulus properties we obtain some new general inequalities of the Hermite-Hadamard and Bullen type for functions whose derivatives in absolute value at certain power are arithmetically harmonically (AH) convex. In the last part of the article, applications including arithmetic mean, geometric mean, harmonic mean, logarithmic mean and p-logarithmic mean, which are some special means of real numbers, are given by using arithmetic harmonically convex functions.

On inequalities of Simpson's type for convex functions via generalized fractional integrals

Fractional calculus and applications have application areas in many different fields such as physics, chemistry, and engineering as well as mathematics. The application of arithmetic carried out in classical analysis in fractional analysis is very important in terms of obtaining more realistic results in the solution of many problems. In this study, we prove an identity involving generalized fractional integrals by using differentiable functions. By utilizing this identity, we obtain several Simpson’s type inequalities for the functions whose derivatives in absolute value are convex. Finally, we present some new results as the special cases of our main results.

Certain midpoint-type integral inequalities involving twice differentiable generalized convex mappings and applications in fractal domain

The conception of extended fractal left- as well as right-side integral operators and the related Hermite–Hadamard’s integral inequalities along with midpoint are proposed firstly in this paper. A midpoint-type integral identity in fractal domain, involving twice differentiable mappings, is then derived based on the introduced extended fractal integral operators for the objective of deriving fractal midpoint-type integral inequalities with the mappings whose second-order derivatives in absolute value are required to be generalized convex. Finally, a series of fractal outcomes regarding ɛ-type special means, moments of random variable and wave equations are acquired as applications, correspondingly.

PROPERTIES AND INTEGRAL INEQUALITIES INVOLVING WITH THE GENERALIZED s-TYPE PREINVEX MAPPINGS IN FRACTAL SPACE

As a generalization of the convex mappings, the generalized [Formula: see text]-type preinvex mappings are firstly introduced. Their meaningful properties are then investigated and the Hermite–Hadamard-type integral inequalities via the newly proposed mappings in fractal space are developed. In accordance with the newly proposed identity with three parameters, it is interesting to present certain integral inequalities with regard to the mappings whose first-order derivatives in absolute value belong to the generalized [Formula: see text]-type preinvexity. As applications, on the basis of local fractional calculus, certain inequalities in view of numerical integration, [Formula: see text]-type special means, as well as moments of random variable, are acquired, respectively.

Estimation of the parameterized integral inequalities involving generalized p-convex mappings on fractal sets and related applications

In this article, the author establishes some new parameterized local fractional integral inequalities for mappings whose first local fractional derivatives in absolute values at some powers are generalized p-convex function. Furthermore, certain applications of main results for α-type special means, trapezoidal formula and cumulative distribution function are used to verify the computational effectiveness and relevance of the presented technique.

New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity

This study provokes the existence of quantum Hermite-Hadamard inequalities under the concept of q-integral. We analyse and illustrate a new identity for the differentiable function mappings whose second derivatives in absolute value are (α,m) convex. Some basic inequalities such as Hölder’s and Power mean have been used to obtain new bounds and it has been determined that the main findings are generalizations of many results that exist in the literature. We make links between our findings and a number of well-known discoveries in the literature. The conclusion in this study unify and generalise previous findings on Hermite-Hadamard inequalities.

On some new generalized fractional inequalities for twice differentiable functions

In this paper, we establish an identity involving Sarikaya fractional integrals for twice differentiable functions. We obtain some new generalized fractional inequalities for the functions whose second derivatives in absolute value are convex by utilizing obtained equality. Utilizing the new inequalities obtained, some new inequalities for Riemann–Liouville fractional integrals and k-Riemann–Liouville fractional integrals are obtained. In addition, some of these results generalize ones obtained in earlier works.

Strongly n-polynomial convexity and related inequalities

"In this paper, we introduce and study the concept of strongly n-polynomial convexity functions and their some algebraic properties. We prove two Hermite-Hadamard type inequalities for the newly intro- duced class of functions. In addition, we obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respec- tively at least one, is strongly n-polynomial convexity. Also, we compare the obtained results with both Hölder, Hölder- Işcan inequalities and power-mean, improved-power-mean integral inequalities and show that the re- sult obtained with H ̈older- ̇Is ̧can and improved power-mean inequalities give better approach than the others."

Certain error bounds on the parameterized integral inequalities in the sense of fractal sets

The objective of this study is to research certain integral inequalities with a parameter through the generalized (s, P)-preinvex mappings in the frame of fractal space. In view of this, we propose and investigate the conception of the generalized (s, P)-preinvex mappings and their related properties. Meanwhile, we establish an integral identity in the settings of fractal sets and present the parameterized integral inequalities for mappings whose first-order derivatives in absolute value belong to the generalized (s, P)-preinvexity. As applications with regard to local fractional integral operators, we consider applying the derived findings to v-type special means, numerical integrations, as well as extended probability distribution mappings, respectively.

Fuzzy Ostrowski type inequalities via ϕ - λ -convex functions

We would like to state well-known Ostrowski inequality via \(\phi\)-\(\lambda\)-convex by using the Fuzzy Reimann integrals. In addition, we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are \(\phi\)-\(\lambda\)-convex by Holder's and power mean inequalities. We are introducing very first time that the class of \(\phi\)-\(\lambda\)-convex function, which is the generalization of many important classes including class of \(h\)-convex, Godunova-Levin \(s\)-convex, \(s\)-convex in the \(2^{\rm nd}\) kind and hence contains convex functions. It also contains class of \(P\)-convex and class of Godunova-Levin. In this way we also capture the results with respect to convexity of functions.

Generalizations of Hermite-Hadamard, Bullen and Simpson inequalities via h−convexity

Abstract In this paper, the authors established a new identity for differentiable functions, afterward, they obtained some new general inequalities for functions whose first derivatives in absolute value at certain powers are h-convex by using the identity. On the other hand, a general inequality is studied, which gives Hermite-Hadamard, Bullen and Simpson inequalities. Also, they gave some applications for special means for arbitrary positive numbers.

PROPERTIES AND INTEGRAL INEQUALITIES ARISING FROM THE GENERALIZED n-POLYNOMIAL CONVEXITY IN THE FRAME OF FRACTAL SPACE

First, we define what we named the generalized [Formula: see text]-polynomial convex mappings as a generalization of convex mappings, investigate their meaningful properties, and establish two Hermite–Hadamard’s-type integral inequalities via the newly proposed mappings in the frame of fractal space as well. Second, in accordance with the discovered identity with a parameter, we present certain improved integral inequalities with regard to the mappings whose first-order derivatives in absolute value belong to the generalized [Formula: see text]-polynomial convexity. As applications, on the basis of local fractional calculus, we acquire three inequalities in view of special means, numerical integrations, as well as probability density mappings, respectively.

AN IMPROVEMENT OF THE POWER-MEAN INTEGRAL INEQUALITY IN THE FRAME OF FRACTAL SPACE AND CERTAIN RELATED MIDPOINT-TYPE INTEGRAL INEQUALITIES

First, we construct a reformative version of the power-mean integral inequality in the sense of fractal space. Second, we define what we named as the generalized [Formula: see text]-convex mappings, and investigate some related properties. Moreover, in accordance with the derived midpoint-type integral identities on fractal space, we establish certain improvements of the midpoint-type integral inequalities for mappings whose first-order derivatives in absolute value belong to the generalized [Formula: see text]-convex mappings. As applications in association with local fractional calculus, we acquire three inequalities considering [Formula: see text]-type arithmetic mean and [Formula: see text]-logarithmic mean, numerical integration, as well as probability density mappings, respectively.

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.

New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.</p></abstract>

Fuzzy Ostrowski type inequalities via h-convex

We would like to state well-known Ostrowski inequality via h-convex by using the Fuzzy Reimann integrals. In addition, we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are h-convex by Holder’s and power mean inequalities. This class of h-convex function, which is the generalization of many important classes including class of Godunova-Levin s-convex, s-convex in the 2nd kind and hence contains convex functions. It also contains class of P-convex and class of Godunova-Levin. In this way we also capture the results with respect to convexity of functions.

New inequalities of hermite-hadamard type for functions whose second derivatives absolute values are exponential trigonometric convex

New inequalities of hermite-hadamard type for functions whose second derivatives absolute values are exponential trigonometric convex

Hermite-Hadamard Type Inequalities Related to s-Convex Functions in the Third Sense

In this paper, some Hermite-Hadamard type inequalites for $s$-convex functions in the third sense are studied. It is established several new inequalities for functions whose derivative in absolute value and $p$th power of its derivative in absolute value are $s$-convex in the third sense. In addition, these inequalities are used to find an upper bound for error in numerical integration for this type of functions.

New Hermite-Hadamard type inequalities for m and (α, m)-convex functions on the coordinates via generalized fractional integrals

In this paper, we obtained a new Hermite-Hadamard type inequality for functions of two independent variables that are m-convex on the coordinates via some generalized Katugampola type fractional integrals. We also established a new identity involving the second order mixed partial derivatives of functions of two independent variables via the generalized Katugampola fractional integrals. Using the identity, we established some new Hermite-Hadamard type inequalities for functions whose second order mixed partial derivatives in absolute value at some powers are (α, m)-convex on the coordinates. Our results are extensions of some earlier results in the literature for functions of two variables.

Fuzzy Ostrowski type inequalities via h-convex

We would like to state well-known Ostrowski inequality via h-convex by using the Fuzzy Reimann integrals. In addition, we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are h-convex by Holder’s and power mean inequalities. This class of h-convex function, which is the generalization of many important classes including class of Godunova-Levin s-convex, s-convex in the 2nd kind and hence contains convex functions. It also contains class of P-convex and class of Godunova-Levin. In this way we also capture the results with respect to convexity of functions.