In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is actually false; however, the statement is true for acute triangles, as shown by F. Balitrand in 1916.

Statement of the inequality

Consider an acute triangle (meaning a triangle with three acute angles) in the Euclidean plane with side lengths a, b and c and area S. Then This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides 1,1,1 and area \sqrt{3}/4.

Proof

Dividing both sides of the inequality by 64(abc)^4, we obtain: Using the formula for the area of triangle, and applying the cosines law to the left side, we get: And then using the identity which is true for all triangles in euclidean plane, we transform the inequality above into: Since the angles of the triangle are acute, the tangent of each corner is positive, which means that the inequality above is correct by AM-GM inequality.