[Paper Review] A generalization of Ostrowski type inequality for mappings whose second derivatives belong to L$_{1}\left(a,b ight) $ and applications

Ather Qayyum, Ibrahima Faye+2 more|arXiv (Cornell University)|Mar 28, 2015

Mathematical Inequalities and Applications7 references3 citations

TL;DR

This paper generalizes Ostrowski-type inequalities for functions whose second derivatives are in $L_1(a,b)$, introducing a new inequality involving a parameter $h$ that unifies and extends previous results. The key contribution is a sharp error bound for numerical integration using the trapezoid and midpoint rules, with applications to special means and improved remainder estimates.

ABSTRACT

In this paper, we will improve and generalize inequality of Ostrowski type for mappings whose second derivatives belong to L$_{1}\left(a,b ight) $ . Some well known inequalities can be derived as special cases. In addition, perturbed mid-point inequality and perturbed trapezoid inequality are also obtained. The obtained inequalities have immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid and midpoint formula. Applications to some special means are also investigated.

Motivation & Objective

• To generalize existing Ostrowski-type inequalities by extending them to functions with second derivatives in $L_1(a,b)$, rather than requiring bounded first derivatives.
• To derive a unified inequality involving a parameter $h$ that interpolates between midpoint and trapezoid rules.
• To establish perturbed versions of the midpoint and trapezoid rules with explicit error bounds in terms of $\|f''\|_1$.
• To apply the new inequality to derive new estimates for special means such as power mean, harmonic mean, and logarithmic mean.
• To improve error estimates in numerical integration by providing tighter remainder bounds for composite quadrature rules.

Proposed method

• Derive a new Ostrowski-type inequality using integration by parts and properties of $L_1$-normed second derivatives.
• Introduce a parameter $h \in [0,1]$ to interpolate between the standard midpoint and trapezoid rules, allowing for a generalized error expression.
• Use the identity $f(x) = \frac{1}{b-a}\int_a^b f(t)dt + \text{error term}$ and bound the error via $\|f''\|_1$.
• Apply the generalized inequality to specific mappings ($f(x) = x^r$, $f(x) = 1/x$, $f(x) = \ln x$) to derive inequalities for special means.
• Establish perturbed midpoint and trapezoid rules by modifying the standard formulas with correction terms involving $f'(a)$, $f'(b)$, and $h$.
• Derive composite error bounds for $n$-segment quadrature rules using the local error estimate $\frac{h_i^2}{8}\|f''\|_1$.

Experimental results

Research questions

• RQ1 How can Ostrowski-type inequalities be generalized for functions with second derivatives in $L_1(a,b)$, rather than requiring bounded first derivatives?
• RQ2 What is the optimal error bound for the perturbed midpoint and trapezoid rules when $f'' \in L_1(a,b)$?
• RQ3 Can the generalized inequality be used to derive new inequalities for special means such as the arithmetic, geometric, harmonic, and logarithmic means?
• RQ4 How do the new error bounds compare to classical ones in terms of sharpness and applicability in numerical integration?
• RQ5 What is the role of the parameter $h$ in balancing the contributions of the midpoint and trapezoid rules in the generalized inequality?

Key findings

• The paper establishes a new generalized Ostrowski-type inequality for $f \in C^2[a,b]$ with $f'' \in L_1(a,b)$, valid for $x \in [a + h\frac{b-a}{2}, b - h\frac{b-a}{2}]$, with error bound proportional to $\left[\frac{1}{2}(b-a)(1-h) + |x - \frac{a+b}{2}|\right]^2 \|f''\|_1$.
• For the perturbed trapezoid rule, the remainder term satisfies $|R_T| \leq \sum_{i=0}^{n-1} \frac{h_i^2}{8}\|f''\|_1$, improving classical estimates.
• For the perturbed midpoint rule, the remainder is bounded by $\frac{1}{8}(b-a)^2(1-h)^2\|f''\|_1$, which reduces to $\frac{1}{8}(b-a)^2\|f''\|_1$ when $h=0$.
• When applied to $f(x) = x^r$, the inequality yields $|A^r - L_r^r(a,b)| \leq \frac{1}{8}(b-a)^2 |r(r-1)L_{r-1}^{r-1}(a,b)|$, providing a new bound for power means.
• For $f(x) = \frac{1}{x}$, the inequality yields $|A^{-1} - L^{-1}(a,b)| \leq \frac{1}{4}(b-a)^2 L_{-3}^{-3}(a,b)$, a new estimate for the harmonic mean.
• For $f(x) = \ln x$, the inequality leads to $|\ln A - \ln I(a,b)| \leq \frac{1}{8}(b-a)^2 L_{-2}^{-2}(a,b)$, implying $\left|\frac{A}{I}\right| \leq \exp\left(\frac{(b-a)^2}{8}L_{-2}^{-2}(a,b)\right)$.