(untitled) (European Journal of Pure and Applied Mathematics, 2026)
Let a, b, c be the side lengths of an acute triangle in the Euclidean plane, and let Δ denote its area. The classical inequality of Ono asserts that \[ 27 (b^2+c^2-a^2)^2(c^2+a^2-b^2)^2(a^2+b^2-c^2)^2 (4Δ)^6, \] 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 α > 0 and β > 0, \[ (4Δ)^2α+β C_α,β (b^2+c^2-a^2)^α (c^2+a^2-b^2)^α (a^2+b^2-c^2)^β, \] where the optimal constant C_α,β is computed explicitly. The classical Ono inequality is recovered as the special case α=β=2. We also present an algebraic reformulation of the result and discuss the geometric structure underlying the equality cases.