Parameterized Fractional Newton-Type Inequalities for Ï,h Convex Functions via Generalized Riemann–Liouville Operators With Applications

Liqiang Chen and Norazrizal Aswad Abdul Rahman

Journal of Mathematics, 2026, vol. 2026, 1-13

Abstract: Motivated by the recent work of Samraiz, Ashraf and Naheed [J. Inequal. Appl. (2026)], which established Newton-type inequalities for differentiable p-convex functions via k,s-Riemann–Liouville fractional integrals, the present paper develops a parameterized framework for related Newton-type estimates. Two ingredients are considered. First, a kernel parameter vector λ=λ1,λ2,λ3∈0,13 that simultaneously controls the three subinterval contributions of a Newton–Cotes 3/8 partition; choosing λ=5/8,1/2,3/8 recovers the classical Newton–Cotes 3/8 kernel used in the cited paper, while other choices yield a continuous family of estimators. Second, we use an operator-compatible notion of Ï ,h-convexity, where Ï =s+1 is the coordinate exponent induced by the k,s-Riemann–Liouville kernel. This notion includes ordinary convexity, h-convexity, s-convexity in the second sense, and the compatible positive-order p-convex case as special choices. We first establish a master parameterized fractional integral identity that underlies all the subsequent inequalities. Building on this identity, we prove three families of Newton-type inequalities for absolutely continuous functions under Ï ,h-convexity assumptions on Φ′ or Φ′q: a basic absolute-value inequality, a Hölder-type refinement, and a power-mean-type refinement. We further prove a fractional Newton-type inequality for functions of bounded variation, requiring no differentiability assumption. Several known fixed-kernel inequalities are obtained as special cases in the compatible parameter regime; the explicit reductions are stated. Departing from the cited paper, we present applications to the identric and Stolarsky means, together with a quadrature estimate for the exponential function. Numerical comparison also illustrates that the power-mean refinement is sharper than the Hölder one for the specific examples considered. The results yield explicit, parameter-dependent error bounds for the Newton–Cotes 3/8 quadrature rule and concrete estimates for the identric and Stolarsky means. We close with a discussion of how the framework extends to quantum, multiplicative, and Katugampola calculi.MSC2020: 26A33, 26A51, 26D10, 26D15, 41A55

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:2085205

DOI: 10.1155/jom/2085205

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