Parameterized Fractional Newton‐Type Inequalities for ( ρ , h ) Convex Functions via Generalized Riemann–Liouville Operators With Applications

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 = ( λ ₁ , λ ₂ , λ ₃ ) ∈ [0, 1] ³ 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 ( ρ , h )‐convexity, where ρ = 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 ( ρ , 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