Ono's inequality

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles, as shown by Balitrand in 1916.

Statement of the inequality[edit]

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

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

This inequality fails for general triangles (which was Ono's original conjecture), as shown by the counterexample a = 3/4, b = 1/2, c = 1.

External links[edit]

References[edit]

  • Balitrand, F. (1916). "Problem 4417". Intermed. Math. 23: 86–87. 
  • Ono, T. (1914). "Problem 4417". Intermed. Math. 21: 146. 
  • Quijano, G. (1915). "Problem 4417". Intermed. Math. 22: 66.