Several Integral Inequalities of Hermite–Hadamard Type Related to k-Fractional Conformable Integral Operators

Muhammad Tariq; Soubhagya Kumar Sahoo; Hijaz Ahmad; Thanin Sitthiwirattham; Jarunee Soontharanon

doi:10.3390/sym13101880

Several Integral Inequalities of Hermite–Hadamard Type Related to k-Fractional Conformable Integral Operators

Symmetry ◽

10.3390/sym13101880 ◽

2021 ◽

Vol 13 (10) ◽

pp. 1880

Author(s):

Muhammad Tariq ◽

Soubhagya Kumar Sahoo ◽

Hijaz Ahmad ◽

Thanin Sitthiwirattham ◽

Jarunee Soontharanon

Keyword(s):

Convex Functions ◽

Convex Sets ◽

Convex Geometry ◽

Integral Operators ◽

Symmetric Convex ◽

Strong Correlations ◽

Power Mean ◽

First Order ◽

Mathematical Sciences ◽

Power Mean Inequality

In this paper, we present some ideas and concepts related to the k-fractional conformable integral operator for convex functions. First, we present a new integral identity correlated with the k-fractional conformable operator for the first-order derivative of a given function. Employing this new identity, the authors have proved some generalized inequalities of Hermite–Hadamard type via Hölder’s inequality and the power mean inequality. Inequalities have a strong correlation with convex and symmetric convex functions. There exist expansive properties and strong correlations between the symmetric function and various areas of convexity, including convex functions, probability theory, and convex geometry on convex sets because of their fascinating properties in the mathematical sciences. The results of this paper show that the methodology can be directly applied and is computationally easy to use and exact.

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Some new inequalities for convex functions via Riemann-Liouville fractional integrals

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2020.2.00015 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mustafa Gürbüz ◽

Çetin Yıldız

Keyword(s):

Fractional Integral ◽

Integral Operators ◽

Fractional Integrals ◽

Power Mean ◽

Semigroup Property ◽

Fractional Analysis ◽

Power Mean Inequality ◽

Fractional Orders ◽

Scientific Fields ◽

New Inequalities

AbstractFractional analysis has evolved considerably over the last decades and has become popular in many technical and scientific fields. Many integral operators which ables us to integrate from fractional orders has been generated. Each of them provides different properties such as semigroup property, singularity problems etc. In this paper, firstly, we obtained a new kernel, then some new integral inequalities which are valid for integrals of fractional orders by using Riemann-Liouville fractional integral. To do this, we used some well-known inequalities such as Hölder's inequality or power mean inequality. Our results generalize some inequalities exist in the literature.

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Hermite–Hadamard-type inequalities for geometrically r-convex functions in terms of Stolarsky’s mean with applications to means

Advances in Difference Equations ◽

10.1186/s13662-021-03517-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Muhammad Amer Latif

Keyword(s):

Integral Inequality ◽

Convex Functions ◽

Power Mean ◽

Power Mean Inequality

AbstractIn this paper, we obtain new Hermite–Hadamard-type inequalities for r-convex and geometrically convex functions and, additionally, some new Hermite–Hadamard-type inequalities by using the Hölder–İşcan integral inequality and an improved power-mean inequality.

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Symmetric convex sets with minimal Gaussian surface area

American Journal of Mathematics ◽

10.1353/ajm.2021.0000 ◽

2021 ◽

Vol 143 (1) ◽

pp. 53-94

Author(s):

Steven Heilman

Keyword(s):

Surface Area ◽

Convex Sets ◽

Symmetric Convex ◽

Gaussian Surface

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Simpson type inequalities and applications

The Journal of Analysis ◽

10.1007/s41478-021-00319-4 ◽

2021 ◽

Author(s):

Muhammad Uzair Awan ◽

Muhammad Zakria Javed ◽

Michael Th. Rassias ◽

Muhammad Aslam Noor ◽

Khalida Inayat Noor

Keyword(s):

Quadrature Formula ◽

Convex Functions ◽

Integral Identity ◽

Higher Order ◽

Auxiliary Result ◽

First Order ◽

New Class ◽

Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

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Decomposable Convexities in Graphs and Hypergraphs

ISRN Combinatorics ◽

10.1155/2013/453808 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Francesco M. Malvestuto

Keyword(s):

Convex Hull ◽

Convex Sets ◽

Nonempty Subset ◽

Convex Set ◽

Convex Geometry ◽

Convexity Space ◽

Vertex Set ◽

Convex Geometries ◽

Acyclic Hypergraph ◽

Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

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The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

Symmetry ◽

10.3390/sym10090380 ◽

2018 ◽

Vol 10 (9) ◽

pp. 380 ◽

Cited By ~ 2

Author(s):

Yongtao Li ◽

Xian-Ming Gu ◽

Jianxing Zhao

Keyword(s):

Geometric Mean ◽

Arithmetic Mean ◽

Hölder Inequality ◽

Power Mean ◽

Weighted Arithmetic ◽

Mathematical Equivalence ◽

Power Mean Inequality ◽

Mathematical Relations

In the current note, we investigate the mathematical relations among the weighted arithmetic mean–geometric mean (AM–GM) inequality, the Hölder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power-mean inequality and the Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

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