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 Takashi Ono 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 ≤ ( 4 S ) 6 . {\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 a = 2 , b = 3 , c = 4 , S = 3 15 / 4. {\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 1 , 1 , 1 {\displaystyle 1,1,1} and area 3 / 4. {\displaystyle {\sqrt {3}}/4.}

Proof

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

27 ( b 2 + c 2 − a 2 ) 2 4 b 2 c 2 ( c 2 + a 2 − b 2 ) 2 4 a 2 c 2 ( a 2 + b 2 − c 2 ) 2 4 a 2 b 2 ≤ 4 S 2 b 2 c 2 4 S 2 a 2 c 2 4 S 2 a 2 b 2 {\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 S = 1 2 b c sin ⁡ A {\displaystyle S={\tfrac {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 ≤ ( sin ⁡ A sin ⁡ B sin ⁡ C ) 2 {\displaystyle 27(\cos {A}\cos {B}\cos {C})^{2}\leq (\sin {A}\sin {B}\sin {C})^{2}}

And then using the identity tan ⁡ A + tan ⁡ B + tan ⁡ C = tan ⁡ A tan ⁡ B tan ⁡ C {\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:

27 ( tan ⁡ A tan ⁡ B tan ⁡ C ) ≤ ( tan ⁡ A + tan ⁡ B + tan ⁡ C ) 3 {\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

References

  • 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.

External links

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