a,b,c are positive real numbers with abc=1 then prove that (a / a+b⁴+c⁴) + (b / a⁴+b+c⁴) + (c / a⁴+b⁴+c)＜=1

Question

a,b,c are positive real numbers

abc=1

prove that (a / a+b⁴+c⁴) + (b / a⁴+b+c⁴) + (c / a⁴+b⁴+c)＜=1

Answer 1

Naively,

a = 1/bc < 1

b = 1/ac < 1

c = 1/ab < 1

then a^4 < 1, b^4 < 1, c^4 < 1

so a + b^4 + c^4 < 1 + 1+1 = 3

a^4 + b^4 + c < 1+1+1 = 3

a^4 + b + c^4 < 1 +1+1 = 3

So each of those denominators are no more than three.

Most importantly, 0 < a < 1 means 0 < a^2 < a < 1 by multiplying everything by a.

0 < a^3 < a^2 < a < 1 by multiplying everything by a.

0 < a^4 < a^3 < a^2 < a < 1 by multiplying everything by a.

The same holds for b and c.

so the fractons in the statement of the problem are PROPER fraction and are less than 1/3 + 1/3+ 1/3 = 1

end of proof