Ono's inequality

Theorem about triangles

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

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

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

The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides ${\displaystyle 1,1,1}$ and area ${\displaystyle {\sqrt {3}}/4.}$

Proof

Dividing both sides of the inequality by ${\displaystyle 64(abc)^{4}}$, we obtain:

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

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

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

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

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

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

• Balitrand, F. (1916). "Problem 4417". Interméd. Math. 23: 86–87. JFM 46.0859.06.
• Ono, T. (1914). "Problem 4417". Interméd. Math. 21: 146.
• Quijano, G. (1915). "Problem 4417". Interméd. Math. 22: 66.
• Lukarevski, M. (2017). "An alternate proof of Gerretsen's inequalities". Elem. Math. 72: 2–8.

External links

• Weisstein, Eric W. "Ono inequality". MathWorld.