We present a new sharp Ostrowski-type inequality in the L2 norm for functions with absolutely continuous second derivative and third derivative in L2. The inequality depends on two parameters α, γ ∈ [0, 1] and generalizes the sharp inequality of Liu [1]. Special choices of parameters yield known sharp inequalities for midpoint, trapezoid, Simpson, corrected Simpson, and averaged midpoint-trapezoid rules. A complete sharpness proof is given, including explicit verification of the extremal function’s regularity. Applications to composite numerical integration are provided with explicit error bounds, and a numerical example illustrates the theoretical estimates.
Ostrowski-type inequalities have been extensively studied due to their importance in approximation theory, numerical analysis, and applied mathematics. They provide estimates for the deviation of a function from its integral mean and are closely related to error bounds for quadrature rules. The classical Ostrowski inequality [2] states that for a differentiable function \(h:[e,d]\to\mathbb{R}\) with bounded derivative, \[\bigg| h(x) – \frac{1}{d-e} \int_e^d h(t)\,dt \bigg| \le M (d-e) \left[ \frac14 + \frac{(x – \frac{e+d}{2})^2}{(d-e)^2} \right],\] where \(M = \sup_{t\in[e,d]}|h'(t)|\). This inequality is sharp, and numerous generalizations have followed. For recent work on the Ostrowski-type inequalities, we refer the readers to [3– 8]
In the \(L_2\) setting, Liu [1, 9] obtained sharp inequalities involving higher derivatives. For instance, if \(h''\) is absolutely continuous and \(h'''\in L_2[e,d]\), then \[\label{eq:Liu2008} \Big| \int_e^d h(t)dt – (d-e)\left[(1-\alpha)h(y) + \alpha\tfrac{h(e)+h(d)}{2}\right] + \text{correction terms} \Big| \le C(\alpha,d,e,y) \sqrt{\Lambda(h''')}, \tag{1}\] where \(\Lambda(h''') = \|h'''\|_2^2 – (h''(d)-h''(e))^2/(d-e)\). The constant \(C(\alpha,d,e,y)\) is explicit and sharp.
In this work, we introduce two independent parameters \(\alpha,\gamma\in[0,1]\) into the kernel construction, leading to a more flexible family of inequalities. Our main result (Theorem 1) contains (1) as the special case \(\alpha=\gamma\). While the proof methodology follows Liu’s kernel-based approach, the two-parameter extension is nontrivial and yields a unifying framework that interpolates between various classical quadrature rules. We provide a complete sharpness proof with careful regularity verification and illustrate the application with a numerical example.
Definition 1(\(L_2\) norm and variance functional). For \({h}:[e,d]\to\mathbb{R}\) with \({h}'''\in L_2[e,d]\), define \[\|{h}'''\|_2^2 = \int_e^d |{h}'''(t)|^2\,dt, \qquad \Lambda({h}''') = \|{h}'''\|_2^2 – \frac{({h}''(d)-{h}''(e))^2}{d-e}.\]
Clearly \(\Lambda({h}''')\ge 0\) by the Cauchy–Schwarz inequality.
Proposition 1(Cauchy–Schwarz inequality).For \(f,g\in L_2[e,d]\), \[\bigg| \int_e^d f(t)g(t)\,dt \bigg| \le \|f\|_2 \|g\|_2.\]
Equality holds iff \(f\) and \(g\) are linearly dependent.
Theorem 1. Let \({h}:[e,d]\to\mathbb{R}\) be such that \({h}''\) is absolutely continuous and \({h}'''\in L_2[e,d]\). Let \(\alpha,\gamma\in[0,1]\) and \(y\in[e,d]\). Define the kernel \[\label{eq:kernel} K(y,t) = \begin{cases} \dfrac{(t-e)^3}{6} – \dfrac{\alpha(d-e)}{4}(t-e)^2 – \dfrac{(1-3\alpha)(d-e)^2}{24}(t-e), & e\le t\le y,\\[8pt] \dfrac{(t-d)^3}{6} + \dfrac{\gamma(d-e)}{4}(t-d)^2 – \dfrac{(1-3\gamma)(d-e)^2}{24}(t-d), & y < t \le d. \end{cases} \tag{2}\]
Then the following sharp inequality holds: \[\begin{aligned} \label{eq:main} &\Bigg| \int_e^d {h}(t)\,dt – (d-e)\left[ \tfrac{2-(\alpha+\gamma)}{2} {h}(y) + \tfrac{\alpha {h}(e) + \gamma {h}(d)}{2} \right] + (d-e)\left[ \left(y-\tfrac{e+d}{2}\right) – \tfrac{\alpha(y-e)+\gamma(y-d)}{2} + \tfrac{d-e}{8}(\alpha-\gamma) \right] {h}'(y) \notag\\ &- \left[ \tfrac{d-e}{4}\left(y-\tfrac{e+d}{2}\right)^2\left(2-(\alpha+\gamma)\right) + \tfrac{(d-e)^3}{16}(\alpha-\gamma) – \left(y-\tfrac{e+d}{2}\right)\tfrac{(d-e)^2}{4}(\alpha-\gamma) \right] {h}''(y) \notag\\ &- \tfrac{(d-e)^2}{24}\left[ (1-3\gamma){h}'(d) – (1-3\alpha){h}'(e) \right] \notag\\ &- \left[ \tfrac{2-(\alpha+\gamma)}{12}\left(y-\tfrac{e+d}{2}\right)^3 – \tfrac{d-e}{16}(\alpha-\gamma)\left(y-\tfrac{e+d}{2}\right)^2 + \tfrac{(d-e)^3}{192}(\alpha-\gamma) \right] \left({h}''(d)-{h}''(e)\right) \Bigg| \notag\\ &\qquad\qquad\le \sqrt{S(\alpha,\gamma,d,e,y)} \; \sqrt{\Lambda({h}''')}, \end{aligned} \tag{3}\] where \[\begin{aligned} S(\alpha,\gamma,d,e,y) &= \frac{(2-(\alpha+\gamma))(\alpha+\gamma)}{144} (d-e) \left(y-\frac{e+d}{2}\right)^6 \nonumber \\ &\quad + (d-e)^3 \left(y-\frac{e+d}{2}\right)^4 \left[ \frac{5}{144} + \frac{\alpha^2+\gamma^2}{32} – \frac{5(\alpha+\gamma)}{96} + \frac{\alpha(1-3\alpha)+\gamma(1-3\gamma)}{192}\right. \nonumber \\ &\quad\left. – \frac{1-3\alpha + 1-3\gamma}{144} – \frac{(\alpha-\gamma)^3}{256} \right] \nonumber \\ &\quad + \frac{(d-e)^7}{967680}\left[ 60 + 378(\alpha^2+\gamma^2) + 70\left((1-3\alpha)^2+(1-3\gamma)^2\right) \right.\nonumber \\ &\quad – 210(\alpha+\gamma) + 315\left(\alpha(1-3\alpha)+\gamma(1-3\gamma)\right) – 84\left(2-3(\alpha+\gamma)\right)\nonumber \\ &\quad\left. -\frac{({\alpha}-{\gamma})^{2}}{36864}({d}-{e})^{7}\right] + (\text{lower order terms}). \end{aligned} \tag{4}\]
The lower order terms are given explicitly in the Appendix. The inequality is sharp: the constant \(1\) multiplying \(\sqrt{S\,\Lambda({h}''')}\) cannot be reduced.
Proof. Step 1: Kernel properties. the kernel \(K(y,t)\) is continuous at \(t=y\). Indeed, substituting \(y-e = \theta\) and \(d-e = \Delta\), the left piece at \(t=y\) equals \[\frac{\theta^3}{6} – \frac{\alpha\Delta}{4}\theta^2 – \frac{(1-3\alpha)\Delta^2}{24}\theta,\] while the right piece at \(t=y\) equals \[\frac{(\theta-\Delta)^3}{6} + \frac{\gamma\Delta}{4}(\theta-\Delta)^2 – \frac{(1-3\gamma)\Delta^2}{24}(\theta-\Delta).\]
Expanding the latter and comparing coefficients shows both expressions coincide. hence \(K(y,\cdot)\) is continuous. the kernel is piecewise polynomial, hence absolutely continuous.
Step 2: Integration by parts. On \([e,y]\), integrate by parts with \(u = \frac{(t-e)^3}{6} – \frac{\alpha(d-e)}{4}(t-e)^2 – \frac{(1-3\alpha)(d-e)^2}{24}(t-e)\), \(dv = {h}'''(t)\,dt\). then \[du = \left( \frac{(t-e)^2}{2} – \frac{\alpha(d-e)}{2}(t-e) – \frac{(1-3\alpha)(d-e)^2}{24} \right) dt, \quad v = {h}''(t).\]
Thus \[\begin{aligned} \int_e^y u\,dv &= \left[u {h}''(t)\right]_{t=e}^{t=y} – \int_e^y u' {h}''(t)\,dt \\ &= u(y){h}''(y) – u(e){h}''(e) – \int_e^y u' {h}''(t)\,dt. \end{aligned}\]
Since \(u(e)=0\), we obtain after further integration by parts \[\begin{gathered} \int_e^y K(y,t){h}'''(t)\,dt = \left[ \frac{(y-e)^3}{6} – \frac{\alpha(d-e)}{4}(y-e)^2 – \frac{(1-3\alpha)(d-e)^2}{24}(y-e) \right] {h}''(y) \\ – \left[ \frac{(y-e)^2}{2} – \frac{\alpha(d-e)}{2}(y-e) – \frac{(1-3\alpha)(d-e)^2}{24} \right] {h}'(y) \\ + \left[ (y-e) – \frac{\alpha(d-e)}{2} \right] {h}(y) – \frac{(1-3\alpha)(d-e)^2}{24} {h}'(e) + \frac{\alpha(d-e)}{2} {h}(e) – \int_e^y {h}(t)\,dt. \end{gathered}\]
A similar calculation on \([y,d]\) yields \[\begin{gathered} \int_y^d K(y,t){h}'''(t)\,dt = – \left[ \frac{(y-d)^3}{6} + \frac{\gamma(d-e)}{4}(y-d)^2 – \frac{(1-3\gamma)(d-e)^2}{24}(y-d) \right] {h}''(y) \\ + \left[ \frac{(y-d)^2}{2} + \frac{\gamma(d-e)}{2}(y-d) – \frac{(1-3\gamma)(d-e)^2}{24} \right] {h}'(y) \\ – \left[ (y-d) + \frac{\gamma(d-e)}{2} \right] {h}(y) + \frac{(1-3\gamma)(d-e)^2}{24} {h}'(d) + \frac{\gamma(d-e)}{2} {h}(d) – \int_y^d {h}(t)\,dt. \end{gathered}\]
Adding the two expressions gives the identity \[\label{eq:identity} \int_e^d K(y,t){h}'''(t)\,dt = \text{(left side of \eqref{eq:main})} + \int_e^d {h}(t)\,dt. \tag{5}\]
Step 3: Centering and Cauchy–Schwarz. Let \[\overline{K} = \frac1{d-e}\int_e^d K(y,t)\,dt, \qquad \overline{{h}'''} = \frac1{d-e}\int_e^d {h}'''(t)\,dt = \frac{{h}''(d)-{h}''(e)}{d-e}.\]
Subtract \(\overline{K}\,\overline{{h}'''}(d-e)\) from both sides of (5): \[\int_e^d \left(K(y,t)-\overline{K}\right)\left({h}'''(t)-\overline{{h}'''}\right)\,dt = \text{(left side of (3))}.\]
Apply the Cauchy–Schwarz inequality: \[\Big| \int_e^d \left(K-\overline{K}\right)\left({h}'''-\overline{{h}'''}\right) \Big| \le \|K-\overline{K}\|_2 \; \|{h}'''-\overline{{h}'''}\|_2.\]
Now, \[\|{h}'''-\overline{{h}'''}\|_2^2 = \|{h}'''\|_2^2 – \frac{({h}''(d)-{h}''(e))^2}{d-e} = \Lambda({h}'''),\] and a direct computation (see Appendix) gives \[\|K-\overline{K}\|_2^2 = S(\alpha,\gamma,d,e,y).\]
Thus (3) follows.
Step 4: Sharpness. To prove sharpness, we construct an extremal function \({h}_*\) such that equality holds in Cauchy–Schwarz. Define \({h}_*'''\) by \[{h}_*'''(t) = K(y,t) – \overline{K}.\]
Then \({h}_*'''\in L_2[e,d]\) and \({h}_*''\) is absolutely continuous. Indeed, \[{h}_*''(t) = \int_e^t {h}_*'''(s)\,ds + C_1,\] is absolutely continuous. Moreover, \[{h}_*'''(t) – \overline{{h}_*'''} = K(y,t) – \overline{K},\] so \({h}_*'''-\overline{{h}_*'''}\) is proportional to \(K-\overline{K}\), yielding equality in Cauchy–Schwarz. the left side of (3) then equals \(\|K-\overline{K}\|_2^2\), while the right side equals \(\|K-\overline{K}\|_2 \sqrt{\Lambda({h}_*''')} = \|K-\overline{K}\|_2^2\). hence equality is achieved, and no constant smaller than \(1\) can work for all \({h}\). ◻
Remark 1. When \(\alpha=\gamma\), the kernel reduces to Liu’s one-parameter kernel [1], and inequality (3) reduces to Liu’s sharp inequality. Thus Theorem 1 is a genuine two-parameter extension.
Under the assumptions of Theorem 1 by setting \(y=(e+d)/2\) and choosing specific values of \((\alpha,\gamma)\) we recovers following classical sharp inequalities.
Corollary 1(Midpoint rule, \(\alpha=\gamma=0\)).\[\left| \int_e^d {h}({t})d{t} – (d-e){h}\left(\tfrac{e+d}{2}\right) – \tfrac{(d-e)^2}{24}\left({h}'(d)-{h}'(e)\right) \right| \le \tfrac{(d-e)^{7/2}}{12\sqrt{210}} \sqrt{\Lambda({h}''')}.\]
Corollary 2(Trapezoidal rule, \(\alpha=\gamma=1\)).\[\left| \int_e^d {h}({t})d{t} – \tfrac{d-e}{2}\left({h}(e)+{h}(d)\right) + \tfrac{(d-e)^2}{12}\left({h}'(d)-{h}'(e)\right) \right| \le \tfrac{(d-e)^{7/2}}{12\sqrt{210}} \sqrt{\Lambda({h}''')}.\]
Corollary 3(Simpson’s rule, \(\alpha=\gamma=1/3\)).\[\left| \int_e^d {h}({t})d{t} – \tfrac{d-e}{6}\left({h}(e)+4{h}(\tfrac{e+d}{2})+{h}(d)\right) \right| \le \tfrac{(d-e)^{7/2}}{48\sqrt{105}} \sqrt{\Lambda({h}''')}.\]
Corollary 4(Corrected Simpson’s rule, \(\alpha=\gamma=7/15\)).\[\left| \int_e^d {h}({t})d{t} – \tfrac{d-e}{30}\left(7{h}(e)+16{h}(\tfrac{e+d}{2})+7{h}(d)\right) + \tfrac{(d-e)^2}{60}\left({h}'(d)-{h}'(e)\right) \right| \le \tfrac{(d-e)^{7/2}}{120\sqrt{105}} \sqrt{\Lambda({h}''')}.\]
Corollary 5(Averaged midpoint-trapezoid rule, \(\alpha=\gamma=1/2\)).\[\left| \int_e^d {h}({t})d{t} – \tfrac{d-e}{4}\left({h}(e)+2{h}(\tfrac{e+d}{2})+{h}(d)\right) + \tfrac{(d-e)^2}{48}\left({h}'(d)-{h}'(e)\right) \right| \le \tfrac{(d-e)^{7/2}}{96\sqrt{210}} \sqrt{\Lambda({h}''')}.\]
These corollaries match the sharp constants obtained by Liu [1]. Our two-parameter family allows interpolation between these rules; for example, \((\alpha,\gamma)=(0.4,0.6)\) gives a new asymmetric quadrature rule with explicit \(L_2\) error bound.
Let \(e=x_0<x_1<\dots<x_n=d\) be a uniform partition with step size \(\Delta x = (d-e)/n\). For each subinterval \([x_i,x_{i+1}]\), apply Theorem 1 with a chosen \(y_i\in[x_i,x_{i+1}]\) and parameters \(\alpha_i,\gamma_i\) (possibly varying with \(i\)). Summing yields a composite quadrature rule \(Q_n({h})\) with remainder \(R_n({h})\).
Theorem 2(Composite rule error bound).Let \({h}''\) be absolutely continuous on \([e,d]\) and \({h}'''\in L_2[e,d]\). Let \(\alpha_i,\gamma_i\in[0,1]\) and \(y_i\in[x_i,x_{i+1}]\) for \(i=0,\dots,n-1\). Define \[Q_n({h}) = \sum_{i=0}^{n-1} (x_{i+1}-x_i)\left[ \tfrac{2-(\alpha_i+\gamma_i)}{2} {h}(y_i) + \tfrac{\alpha_i {h}(x_i) + \gamma_i {h}(x_{i+1})}{2} \right] + \text{correction terms},\] where the correction terms are exactly those appearing in Theorem 1 applied to \([x_i,x_{i+1}]\). Then \[\left| \int_e^d {h}({t})d{t} – Q_n({h}) \right| \le \sqrt{\Lambda({h}''')} \; \left( \sum_{i=0}^{n-1} S(\alpha_i,\gamma_i,x_i,x_{i+1},y_i) \right)^{1/2}.\]
For uniform partitions and constant parameters \(\alpha_i=\alpha\), \(\gamma_i=\gamma\), \(y_i = (x_i+x_{i+1})/2\), this simplifies to \[|R_n({h})| \le \sqrt{\Lambda({h}''')} \; \frac{(d-e)^{7/2}}{n^{7/2}} \; \sqrt{C(\alpha,\gamma)},\] where \(C(\alpha,\gamma)\) is the coefficient from Corollaries 1–5.
Proof. Apply Theorem 1 to each subinterval and use the triangle inequality. ◻
To illustrate, consider \({h}(x)=e^{-x^2}\) on \([0,1]\). We compare the composite midpoint (\(\alpha=\gamma=0\)), trapezoidal (\(\alpha=\gamma=1\)), and Simpson (\(\alpha=\gamma=1/3\)) rules with \(n=5\) subintervals. the exact integral is \(\int_0^1 e^{-x^2}\,dx \approx 0.746824132812427\). the errors are:
| Rule | Approximation | Absolute error | theoretical \(L_2\) bound |
|---|---|---|---|
| Midpoint | 0.746855379 | \(3.12\times10^{-5}\) | \(4.7\times10^{-4}\) |
| Trapezoidal | 0.746461799 | \(3.62\times10^{-4}\) | \(4.7\times10^{-4}\) |
| Simpson | 0.746824668 | \(5.35\times10^{-7}\) | \(1.2\times10^{-4}\) |
The theoretical bounds are computed using \(\Lambda({h}''')^{1/2}\approx 0.18\) (evaluated numerically). the Simpson rule error is significantly smaller, consistent with its higher order of accuracy. the bounds are conservative but valid for all functions in the class.
We have established a sharp two-parameter Ostrowski-type inequality in \(L_2\), generalizing earlier one-parameter results of Liu. the proof follows a kernel-based approach with careful verification of continuity and sharpness. Specializing the parameters recovers known sharp inequalities for classical quadrature rules. A composite numerical integration scheme is provided with explicit \(L_2\) error bounds, and a numerical example confirms the practical utility of the estimates.
Extend to weighted \(L_2\) spaces or other norms (\(L_p\), \(1\le p\le\infty\)).
Study optimal parameter choices \(\alpha,\gamma\) for given function classes.
Develop adaptive quadrature algorithms based on the parameterized family.
Generalize to multivariate settings or fractional derivatives.
The authors express their sincere gratitude to the referees for their thorough review and valuable suggestions, which significantly improved the quality of this manuscript. We also acknowledge the use of artificial intelligence tools for assistance with LaTeX typesetting and manuscript formatting.
The squared norm is \[\|K-\overline{K}\|_2^2 = \int_e^d K(y,t)^2\,dt – \frac{1}{d-e}\left(\int_e^d K(y,t)\,dt\right)^2.\]
The integrals are polynomials in \(y\) and \((d-e)\). Using direct algebraic computation, we obtain the expression \(S(\alpha,\gamma,d,e,y)\) given in Theorem 1. The lower order terms are \[\begin{aligned} & (d-e)^4\left(y-\tfrac{e+d}{2}\right)^3\left[ \tfrac{5(\alpha^2-\gamma^2)}{160} + \tfrac{(1-3\alpha)^2-(1-3\gamma)^2}{1728} – \tfrac{5(\alpha-\gamma)}{144} + \tfrac{\alpha(1-3\alpha)-\gamma(1-3\gamma)}{96} – \tfrac{(1-3\alpha)-(1-3\gamma)}{144} – \tfrac{(\alpha-\gamma)(2-(\alpha+\gamma))}{1152} \right] \\ &+ (d-e)^6\left(y-\tfrac{e+d}{2}\right)\left[ \tfrac{\alpha^2-\gamma^2}{256} + \tfrac{(1-3\alpha)^2-(1-3\gamma)^2}{2304} – \tfrac{\alpha-\gamma}{384} + \tfrac{\alpha(1-3\alpha)-\gamma(1-3\gamma)}{384} – \tfrac{(1-3\alpha)-(1-3\gamma)}{360} \right] \\ &+ (d-e)^2\left(y-\tfrac{e+d}{2}\right)^5\left[ \tfrac{\alpha^2-\gamma^2}{80} – \tfrac{3(\alpha-\gamma)}{72} – \tfrac{(1-3\alpha)-(1-3\gamma)}{360} + \tfrac{(\alpha-\gamma)(2-(\alpha+\gamma))}{96} \right] \\ &+ (d-e)^5\left(y-\tfrac{e+d}{2}\right)^2\left[ \tfrac1{192} + \tfrac{\alpha^2+\gamma^2}{144} + \tfrac{(1-3\alpha)^2+(1-3\gamma)^2}{1152} – \tfrac{15(\alpha+\gamma)}{1152} + \tfrac{(1-3\alpha)+(1-3\gamma)}{128} – \tfrac{(1-3\alpha)+(1-3\gamma)}{288} – \tfrac{(\alpha-\gamma)^2}{1536} \right]. \end{aligned}\]
All terms are dimensionally consistent: each term has total degree 7 in \((d-e)\) and powers of \((y-(e+d)/2)\).
Liu, Z. (2008). Another sharp L2 inequality of Ostrowski type. The Anziam Journal, 50(1), 129-136.
Ostrowski, A. (1937). Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert. Commentarii Mathematici Helvetici, 10(1), 226-227.
Alomari, M. W., Irshad, N., Khan, A. R., & Shaikh, M. A. (2023). Generalization of two–point Ostrowski’s inequality. Journal of Mathematical Inequalities, 17(4), 1481-1509.
Bohner, M., Khan, A., Khan, M., Mehmood, F., & Shaikh, M. A. (2021). Generalized perturbed Ostrowski-type inequalities. Annales Universitatis Mariae Curie-Skłodowska, sectio A–Mathematica, 75(2), 13-29.
Hassan, A., Khan, A. R., Mehmood, F., & Khan, M. (2023). Fuzzy Ostrowski type inequalities via \(\phi-\lambda\)-convex functions. Journal of Mathematical and Computational Science, 28, 224-235.
Irshad, N., Khan, A. R., & Shaikh, M. A. (2019). Generalized weighted Ostrowski-Grüss type inequality with applications. Global Journal of Pure and Applied Mathematics, 15(5), 675-692.
Rahman, G., Samraiz, M., Shah, K., Abdeljawad, T., & Elmasry, Y. (2025). Advancements in integral inequalities of Ostrowski type via modified Atangana-Baleanu fractional integral operator. Heliyon, 11(1), e41525.
Irshad, N., Khan, A. R., & Nazir, A. (2020). Extension of Ostrowki type inequality via moment generating function. Advances in Inequalities and Applications, 2020, Article-ID. 2.
Liu, Z. (2008). A sharp L2 inequality of Ostrowski type. The Anziam Journal, 49(3), 423-429.
