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Abstract
In this paper, the authors derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. Moreover, three new fractional integral identities are given, and on using them as auxiliary results some interesting integral inequalities are found. Finally, in order to show the efficiency of our main results, some applications to special means for different positive real numbers and error estimations for quadrature formulas are obtained.
Highlights
Introduction and preliminaries Computational and FractionalAnalysis are nowadays more and more at the center of mathematics and of other related sciences, either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world
In order to show the efficiency of our main results, some applications to special means for positive different real numbers and error estimations for quadrature formulas will be obtained
We derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions
Summary
We derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions. 2.3 Further results utilizing auxiliary results obtained in the previous subsection, we derive some further generalized fractional trapezium-like inequalities using the class of harmonic convex functions. Proof Using Lemma 2.3, the modulus property, Hölder’s inequality and the harmonic convexity of |Υ |q, we have 1 p p(τ ) dτ. Proof Using Lemma 2.3, the modulus property, the power mean inequality and the harmonic convexity of of |Υ |q, we have. Proof By using Lemma 2.4, the property of modulus, Hölder’s inequality and the harmonic convexity of |Υ |q, we obtain the desired result. Proof By using Lemma 2.4, the property of modulus, the power mean inequality and the convexity of |Υ |q we obtain the desired result. We omit here their proofs and the details are left to the interested reader
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