Ono's inequality

Template:Short description 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

$27(b^2+c^2-a^2{)}^2(c^2+a^2-b^2{)}^2(a^2+b^2-c^2{)}^2\le (4S{)}^6.$

This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample $a=2,b=3,c=4,S=3\sqrt{15}/4.$

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:

$27\frac{(b^2+c^2-a^2{)}^2}{4b^2{c}^2}\frac{(c^2+a^2-b^2{)}^2}{4a^2{c}^2}\frac{(a^2+b^2-c^2{)}^2}{4a^2{b}^2}\le \frac{4S^2}{b^2{c}^2}\frac{4S^2}{a^2{c}^2}\frac{4S^2}{a^2{b}^2}$

Using the formula $S=\frac{1}{2}bc\sin A$ for the area of triangle, and applying the cosines law to the left side, we get:

$27(\cos A\cos B\cos C{)}^2\le (\sin A\sin B\sin C{)}^2$

And then using the identity $\tan A+\tan B+\tan C=\tan A\tan B\tan C$ which is true for all triangles in euclidean plane, we transform the inequality above into:

$27(\tan A\tan B\tan C)\le (\tan A+\tan B+\tan C{)}^3$

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.

See also

• List of triangle inequalities

References

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External links

• Template:MathWorld