[Paper Review] Ostrowski type inequalities for harmonically s-convex functions via fractional integrals

İmdat Işcan|arXiv (Cornell University)|Dec 30, 2013

Mathematical Inequalities and Applications17 references3 citations

TL;DR

This paper establishes a new identity for Riemann–Liouville fractional integrals and derives novel Ostrowski-type inequalities for harmonically s-convex functions. By leveraging Hölder's inequality and properties of the hypergeometric function, it provides sharp bounds involving fractional integrals, generalizing classical Ostrowski inequalities to a broader class of convex functions with improved accuracy through parameterized estimates.

ABSTRACT

In this paper, a new identity for fractional integrals is established. Then by making use of the established identity, some new Ostrowski type inequalities for harmonically s-convex functions via Riemann--Liouville fractional integral are established.

Motivation & Objective

• To establish a new identity for Riemann–Liouville fractional integrals involving harmonically s-convex functions.
• To derive sharp Ostrowski-type inequalities for harmonically s-convex functions using fractional integral operators.
• To generalize classical Ostrowski inequalities by incorporating the s-convexity condition in the harmonic setting.
• To provide explicit bounds involving the hypergeometric function and beta function for improved approximation accuracy.

Proposed method

• A new fractional integral identity is derived using the change of variables and properties of the harmonic mean.
• The proof relies on Hölder’s inequality and the harmonically s-convexity of |f′|^q to bound the fractional integral remainder.
• The bounds are expressed using the hypergeometric function _2F_1 and the beta function β(x,y), enabling precise estimation.
• Parameters α, p, q, s are introduced to generalize the inequality class, with s ∈ (0,1] and q ≥ 1.
• The method applies to differentiability assumptions on f and integrability of f′ on [a,b], ensuring applicability to a wide class of functions.
• The framework allows for reduction to classical cases by setting α = 1 or s = 1, validating consistency with known results.

Experimental results

Research questions

• RQ1 How can Ostrowski-type inequalities be extended to harmonically s-convex functions using fractional calculus?
• RQ2 What role does the hypergeometric function play in refining bounds for fractional integral approximations of harmonically s-convex functions?
• RQ3 Can sharper estimates be obtained by combining Hölder’s inequality with the s-convexity condition in the harmonic setting?
• RQ4 How do the parameters α, p, q, and s influence the tightness of the derived Ostrowski-type inequalities?
• RQ5 What is the relationship between the new fractional integral identity and classical Ostrowski inequalities in the limit cases?

Key findings

• A new fractional integral identity is established that links the difference between f(x) and the weighted integral of f(u)/u² to integrals of f′ involving harmonic means.
• The paper derives two main inequalities (Theorems 11 and 12) that bound the absolute difference using hypergeometric functions and beta functions.
• For |f′(x)| ≤ M, the bound is explicitly given as |S_f(g;α;x,a,b)| ≤ M(1/(αp+1))^(1/p) × {[(x−a)^(α+1)/(ax)^(α−1)] × [λ₁ + λ₂]^(1/q) + [(b−x)^(α+1)/(bx)^(α−1)] × [λ₃ + λ₄]^(1/q)} with λ terms defined via beta and hypergeometric functions.
• When s = 1, the results reduce to known Ostrowski-type inequalities for harmonic convex functions, confirming consistency.
• The bounds are sharp and improve upon classical estimates by incorporating the s-convexity parameter and fractional order α.
• The use of the parameterized fractional integral operator allows for greater flexibility and tighter error estimates in numerical integration approximations.