Volume 1, Issue 1 (2026), Pages 1-16
Some New Opial-Like Inequalities for Two Functions and Applications
Author: Silvestru Sever Dragomir1,2
1Applied Mathematics Research Group, ISILC, Victoria University,
PO Box 14428, Melbourne City, MC 8001, VIC Australia
2Department of Mathematical and Physical Sciences, La Trobe University,
Plenty Road, Bundoora, Melbourne VIC 3086, Australia
Email: sever.dragomir@vu.edu.au
Abstract. In this paper, we establish, among other results, that if \(f,g:[a,b]\to\mathbb{C}\) are absolutely continuous functions satisfying \(f(a)=0\) and \(g(b)=0\), then
\[
\int_{a}^{b}\left| f(t) g(t) \right| \, dt
\leq \frac{1}{4}(b-a)^{2} \int_{a}^{b}\left( \left| f'(t) \right|^{2} + \left| g'(t) \right|^{2} \right) \, dt
– \frac{1}{4} \int_{a}^{b}\left( (t-a)^{2}\left| f'(t) \right|^{2} + (b-t)^{2}\left| g'(t) \right|^{2} \right) \, dt,
\]
provided that the integrals on the right-hand side are finite. In particular, if \(f(a)=f(b)=0\), then the following sharp inequality holds:
\[
\int_{a}^{b} |f(t)|^{2} \, dt \leq \int_{a}^{b} (b-t)(t-a)\, |f'(t)|^{2} \, dt.
\]
We also derive several trapezoid-type and Gruss-type inequalities.
Keywords: Opial’s inequality; trapezoid inequality; Gruss inequality
MSC (2020): 26D15, 26D10
Submitted: December 23, 2025 / Accepted: February 4, 2026/ Published: March 19, 2026
How to cite: S. S. Dragomir, Some new Opial-like inequalities for two functions and applications,
Journal of Advanced Analysis and Applications, 1(1) (2026), 1–16.
This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/