2009/018) Prove that (a+b+c)/3>=3/(1/a+1/b+1/c)? if a,b,c are positive real number.

we know

(a+b+c)/3 >= (abc)^(1/3) AM GM enaquality

(1/a+1/b+1/c)/3 >= (1/(abc))^(`1/3) AM GM enaquality

as both are positive

multiplying

(a+b+c)/3 * (1/1a+1/b+ 1/c)/3 >= 1

or (a+b+c)/3 >= 3/(1/a + 1/b+ 1/c) multiplying both sides by 3/(1/a + 1/b+ 1/c) as this is > 0

proved