已知a>0,b>0,c>0,求证bc/a,+ac/b+ab/c>=a+b+c

Question

已知a>0,b>0,c>0,求证bc/a,+ac/b+ab/c>=a+b+c

Answer 1

证：
bc/a+ac/b+ab/c
=abc/a²+abc/b²+abc/c²
=abc(1/a²+1/b²+1/c²)
(1/a-1/b)²≥0 1/a²+1/b²≥2/ab (1)
(1/b-1/c)²≥0 1/b²+1/c²≥2/bc (2)
(1/a-1/b)²≥0 1/a²+1/c²≥2/ac (3)
(1)+(2)+(3)
2/a²+2/b²+2/c²≥2/ab+2/bc+2/ca
1/a²+1/b²+1/c²≥1/ab+1/bc+1/ca
bc/a+ac/b+ab/c≥abc(1/ab+1/bc+1/ca)=a+b+c
bc/a+ac/b+ab/c≥a+b+c
当a,b,c为正实数时,bc/a+ac/b+ab/c≥a+b+c