[Paper Review] Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions

Wenjun Liu|arXiv (Cornell University)|Oct 15, 2012

Mathematical Inequalities and Applications17 references9 citations

TL;DR

This paper establishes new Ostrowski-type inequalities involving Riemann-Liouville fractional integrals for h-convex functions that are either super-multiplicative or super-additive. By leveraging the h-convexity of the derivative's absolute value and applying power mean and Hölder's inequalities, the authors derive sharp bounds that generalize and extend existing results for s-convex and P-functions, offering tighter estimates in fractional integral approximation contexts.

ABSTRACT

In this paper, some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, which are super-multiplicative or super-additive, are given. These results not only generalize those of Set (2012) and Tunc (2012), but also provide new estimates on these types of Ostrowski inequalities for fractional integrals.

Motivation & Objective

• To extend Ostrowski-type inequalities to the class of h-convex functions using Riemann-Liouville fractional integrals.
• To generalize existing results for s-convex and P-functions by incorporating h-convexity with super-multiplicative or super-additive weight functions.
• To provide tighter error bounds for approximating the integral average of a function via its point evaluation using fractional integral operators.
• To unify and extend recent work on fractional integral inequalities by introducing h-convexity with structural properties on h.

Proposed method

• Utilizes a key identity from Set (2012) relating fractional integrals to derivatives via parameterized integrals over [0,1].
• Applies the power mean inequality and Hölder’s inequality to bound fractional integral expressions involving |f′| when |f′| is h-convex.
• Imposes structural conditions on h: super-multiplicative or super-additive, with h(t) ≥ t for t ∈ [0,1], to derive tighter estimates.
• Uses the definition of h-convexity: f(tx + (1−t)y) ≤ h(t)f(x) + h(1−t)f(y), to bound integrals of |f′| over the interval.
• Derives bounds in terms of Γ-functions and integrals of h(t) and h(1−t), exploiting the homogeneity of the fractional integral kernel.
• Applies the super-multiplicative property h(xy) ≥ h(x)h(y) and super-additive property h(x+y) ≥ h(x)+h(y) to refine estimates in the final inequalities.

Experimental results

Research questions

• RQ1 How can Ostrowski-type inequalities be extended to h-convex functions using Riemann-Liouville fractional integrals?
• RQ2 What are the improved error bounds when the derivative |f′| is h-convex and h is super-multiplicative or super-additive?
• RQ3 How do the new inequalities generalize known results for s-convex and P-functions?
• RQ4 In what way does the super-multiplicative or super-additive nature of h refine the fractional integral estimates compared to standard convexity?
• RQ5 What is the quantitative improvement in the error bound when h(t) = t^s for s ∈ (0,1]?

Key findings

• Theorem 1 provides a fractional Ostrowski inequality for h-convex |f′| with h super-multiplicative and h(t) ≥ t, yielding a bound involving ∫₀¹ [h(t) + h(1−t)] dt.
• Theorem 2 extends the result to the case where |f′| is h-convex and h is super-additive, with the bound depending on ∫₀¹ [h(t) + h(1−t)] dt.
• Theorem 3 establishes a sharp bound under q ≥ 1 and h-super-multiplicativity, with the final inequality involving ∫₀¹ [h(t^{α+1}) + h(t^α(1−t))] dt.
• For h(t) = t^s, the bound reduces to a form matching [24, Theorem 9], confirming consistency with prior work.
• When α = 1, the inequality in Theorem 3 reduces to [25, (2.4)], validating the generalization.
• The bound in Corollary 3 for h(t) = t^s is explicitly quantified using Gamma functions: (α + s + 1)^{-1/q} and Γ(αs + s + 1)/Γ(αs + 1)Γ(s + 1) terms.