Generalization of Ono's Inequality

Abstract

Let $a, b, c$ be the side lengths of an acute triangle in the Euclidean plane, and let $\Delta$ denote its area. The classical inequality of Ono asserts that
\[
27 (b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2} \le (4\Delta)^{6},
\]
with equality if and only if the triangle is equilateral. The traditional proof relies on trigonometric identities and the AM--GM inequality. 

In this paper, we provide a new analytic proof based on a Cartesian coordinate reduction and multivariable optimization techniques. By transforming the geometric inequality into a two-variable polynomial extremal problem, we establish the inequality using elementary tools from multivariable calculus. Furthermore, we derive a nontrivial two-parameter generalization valid for all acute triangles: for any $\alpha > 0$ and $\beta > 0$,
\[
(4\Delta)^{2\alpha+\beta} \ge C_{\alpha,\beta}
(b^{2}+c^{2}-a^{2})^{\alpha}
(c^{2}+a^{2}-b^{2})^{\alpha}
(a^{2}+b^{2}-c^{2})^{\beta},
\]
where the optimal constant $C_{\alpha,\beta}$ is computed explicitly. The classical Ono inequality is recovered as the special case $\alpha=\beta=2$. We also present an algebraic reformulation of the result and discuss the geometric structure underlying the equality cases.