Ono's inequality

Consider an acute or right triangle in the Euclidean plane with side lengths a, b and c and area A. Then

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

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

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 ${\displaystyle {\sqrt {3}}/4.}$

• List of triangle inequalities

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

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