Journal of Advanced Analysis and Applications
Volume 1, Issue 1 (2026), Pages 1–16

Some New Opial-Like Inequalities for Two Functions and Applications
Author: Silvestru Sever Dragomir^1,2

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

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Submitted: December 23, 2025 / Accepted: February 4, 2026/ Published: March 19, 2026

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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.
DOI: https://doi.org/10.66745/jaaa.v1i1.001

This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). https://creativecommons.org/licenses/by/4.0/