Source-linked AI summary
New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals
Erhan Set
TL;DR
The paper addresses Ostrowski-type inequalities for functions whose derivatives satisfy second-sense s-convexity, extending the analysis to fractional integrals. It defines a new identity, uses it to derive Riemann-Liouville fractional-integral inequalities, and relates the results to earlier work by Alomari et al.
Problem
Existing Ostrowski-type inequalities and related s-convexity results motivate extending such estimates to Riemann-Liouville fractional integrals.
Method
The paper introduces an identity for fractional integrals and applies it with assumptions on |f′|, |f′|^q, or s-concavity to derive new inequalities.
Results
The derived results provide new Ostrowski-type estimates for Riemann-Liouville fractional integrals and include a case reducing to an earlier inequality when α = 1.
Takeaways & Limitations
The results extend relationships between Ostrowski inequalities, second-sense s-convexity, and fractional integrals while connecting with Alomari et al.’s findings.
Abstract
from arXiv · showhide
New identity similar to an identity of [13] for fractional integrals have been defined. Then making use of this identity, some new Ostrowski type inequalities for Riemann-Liouville fractional integral have been developed. Our results have some relationships with the results of Alomari et. al., proved in [13] [published in. Appl. Math. Lett. 23 (2010) 1071-1076] and the analysis used in the proofs is simple.
1. Introduction and Preliminary Results
The paper situates new Ostrowski-type bounds within prior integral inequalities, s-convexity results, and fractional-calculus preliminaries. It then proposes new estimates for s-convex functions in the second sense via Riemann-Liouville fractional integrals.
- Ostrowski’s inequality bounds the approximation of an integral average by the value f(x) at a point x ∈ [a, b].
- The paper reviews s-convexity in the second sense, which reduces to ordinary convexity when s = 1.
- Prior work established Hadamard-type and Ostrowski-type inequalities under s-convexity assumptions on f′, |f′|, or |f′|^q.
- The reviewed results include settings involving s-concavity of |f′|^q and bounds |f′(x)| ≤ M on [a, b].
- Fractional-calculus preliminaries are introduced, including the fact that the fractional integral becomes the classical integral when α = 1.
- The paper establishes new Ostrowski-type estimates for second-sense s-convex functions via Riemann-Liouville fractional integrals, motivated by earlier results.
2. Ostrowski Type Inequalities via Fractional Integrals
Using Lemma 2, the paper derives fractional-integral Ostrowski inequalities under s-convexity or s-concavity assumptions on derivatives. Specializing α = 1 recovers earlier inequalities.
- Fractional-integral framework: Lemma 2 provides the basis for deriving the paper’s fractional-integral inequalities.The proofs apply integration by parts, Hölder’s inequality, or the power mean inequality as appropriate.
- s-convex derivative bounds: Theorem 7 establishes a fractional-integral inequality when |f′| is s-convex in the second sense and bounded by M.The result assumes α > 0, s ∈ (0, 1], and f′ ∈ L[a, b].
- Connections to earlier inequalities: Setting α = 1 reduces the fractional inequalities in Theorems 7, 8, and 9 to earlier inequalities from Theorems 3, 4, and 5.The paper explicitly records these reductions as consistency relationships with the earlier results.
- s-convex derivative powers: Theorem 8 extends the result to |f′|^q being s-convex in the second sense for p, q > 1.Its proof uses Lemma 2 together with Hölder’s inequality and the s-convexity bound.
- s-convex derivative powers: Theorem 9 gives another fractional-integral inequality for |f′|^q s-convex in the second sense with q ≥ 1.The proof invokes the power mean inequality and beta- and gamma-function identities.
- s-concave derivative powers: Theorem 10 treats the case where |f′|^q is s-concave in the second sense and derives a corresponding fractional-integral inequality.The proof combines Lemma 2, Hölder’s inequality, and inequalities for s-concave functions.