Minkowski’s inequality for the AB-fractional integral operator

Khan, Hasib; Abdeljawad, Thabet; Tunç, Cemil; Alkhazzan, Abdulwasea; Khan, Aziz

doi:10.1186/s13660-019-2045-3

Cited by 33 publications

(13 citation statements)

References 31 publications

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“…A Gronwall inequality and the Minkowski inequalities via generalized proportional fractional derivative and fractional integral are found in the recent work of Alzabut et al and Rahman et al [6,29]. The Minkowski's inequality for the AB-fractional integral operator is found in [15].…”

Section: Introductionmentioning

confidence: 92%

Some fractional proportional integral inequalities

et al. 2019

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In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n (n ∈ N) positive continuous and decreasing functions on [a, b] are presented. The inequalities presented in this paper are more general than the existing classical inequalities.

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Section: Introductionmentioning

confidence: 92%

Some fractional proportional integral inequalities

et al. 2019

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“…In particular, they have been successfully used in the study of fractional differential equations and sometimes called fractional integral inequalities. They have been vital in proving the uniqueness or existence of solutions for some well-known fractional differential equations and in providing boundedness to solve certain fractional boundary and initial value problems, see for detail [1,7,19,24,26,33,34,41,[48][49][50][51]. The rest of the article is organized as follows: In Sect.…”

Section: Introductionmentioning

confidence: 99%

Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel

Mohammed

Abdeljawad

2020

Adv Differ Equ

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At first, we construct a connection between the Atangana-Baleanu and the Riemann-Liouville fractional integrals of a function with respect to a monotone function with nonsingular kernel. By examining this relationship and the iterated form of Prabhakar fractional model, we are able to find some new Hermite-Hadamard inequalities and related results on integral inequalities for the two models of fractional calculus which are defined using monotone functions with nonsingular kernels.

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“…The concepts of fractional differentiation and fractional integration were examined by Riemann, Liouville, Abel, Laurent, Hardy, and Littlewood. Detailed discussions of fractional calculus and related work can be found in [13][14][15][16]. Fractal analysis is an entirely new field of research based on fractional calculus.…”

Section: Introduction and Prelimnariesmentioning

confidence: 99%

Some new local fractional inequalities associated with generalized $(s,m)$-convex functions and applications

et al. 2020

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Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α to investigate the (s, m)-convexity and relate them to derive generalized Hermite-Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for (s, m)-convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.

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Minkowski’s inequality for the AB-fractional integral operator

Cited by 33 publications

References 31 publications

Some fractional proportional integral inequalities

Some fractional proportional integral inequalities

Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel

Some new local fractional inequalities associated with generalized $(s,m)$-convex functions and applications

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