SOME NEW GENERALIZATIONS OF HADAMARD&amp;ndash;TYPE MIDPOINT INEQUALITIES INVOLVING FRACTIONAL INTEGRALS

“…This theory can be helpful in the subject of statistical issues and quadrature-type formulas. Readers who are interested can refer to [17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

2023

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In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings.

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“…In recent years, many authors (see [7][8][9][10][11][12][13][14][15][16][17][18] and references therein) studied Hermite-Hadamardtype inequalities for improvements and generalizations. In these papers, new inequalities for functions from various convexity classes were obtained.…”

Section: Introductionmentioning

confidence: 99%

Generalization of Hadamard-type trapezoid inequalities for fractional integral operators

Bayraktar

Özdemi̇r

2021

Ufimsk. Mat. Zh.

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The role of convexity theory in applied problems, especially in optimization problems, is well known. The integral Hermite-Hadamard inequality has a special place in this theory since it provides an upper bound for the mean value of a function. In solving applied problems from different fields of science and technology, along with the classical integro-differential calculus, fractional calculus plays an important role. A lot of research is devoted to obtaining an upper bound in the Hermite-Hadamard inequality using operators of fractional calculus.The article formulates and proves the identity with the participation of the fractional integration operator. Based on this identity, new generalized Hadamard-type integral inequalities are obtained for functions for which the second derivatives are convex and take values at intermediate points of the integration interval. These results are obtained using the convexity property of a function and two classical integral inequalities, the Hermite-Hadamard integral inequality and its other form, the power mean inequality. It is shown that the upper limit of the absolute error of inequality decreases in approximately 2 times, where is the number of intermediate points. In a particular case, the obtained estimates are consistent with known estimates in the literature. The results obtained in the article can be used in further researches in the integro-differential fractional calculus.

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SOME NEW GENERALIZATIONS OF HADAMARD–TYPE MIDPOINT INEQUALITIES INVOLVING FRACTIONAL INTEGRALS

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New integral inequalities of Hermite–Hadamard type in a generalized context

New integral inequalities of Hermite–Hadamard type in a generalized context

New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

Generalization of Hadamard-type trapezoid inequalities for fractional integral operators

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