オノの不等式
与式の両辺を 64 ( a b c ) 4 {\displaystyle 64(abc)^{4}} で割る。
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}}}}