# A vague idea

i dont kno and i am not sure if this will be fruitful but i seem to have hit upon a new strategy while i was toying with this 3 variable inequality-
For $a,b,c$ in the interval $[0,1]$ prove that,
$\sum_{cyc}\sqrt {a(1 - b)(1 - c)} \le 1 + \sqrt {abc}$
WHY NOT INCREASE THE VARIABLES?
actually it was like when i saw this inequality i realised that it was pretty trivial with direct AM-GM for the case $c = 0$,
so i was wondering if i could argue like this –
Wishful correct arguement

The LHS of the inequality decreases as one of the variables increase with the other two kept constant and the RHS remains the same.
$\rightarrow$ Implications
LHS will be the greatest when $c = 0$ and then this inequality becomes trivial as i said and we have solved it!!!!

ofcourse the topic “wishful” gave the right hint that i was actually wrong.. for 2 reasons –
1. it is not true with some calculations with the calculator(only thing i am capable of 😦 )
2. It was spotted by my teacher that if we decrease or increase c we get even the RHS to reduce(so obvious but i wasnt a sharp observer like him)

so that turned out to be a good failure but something irked me about the number of variables.. and at the time 2 kept buzzing the corners of my mind..

then with a hunch of wat i saw an year ago i looked up the mixing variables file again by hungkhtn and alas!!! THE VERY BASIC ARGUEMENT WAS THE SAME!! AND NOW I UNDERSTOOD THE METHOD WHICH A YEAR AGO I HAD NO IDEA WAT IT MEANT.. anyway back to the point the file made me think more flexibly on the number of variables..

after a while somehow things took the shape of adding variables..
it wud look like a sort of generalization i thought.. then i remembered the cauchy inductive proof of pham kim hung in the very first page of secrets in inequalities the step of assuming $a_{n + 1} = \frac {Sn}{n}$ and the thoughts lingered..
finally i conjectured the truth of this inequality-
$\sqrt {a(1 - b)(1 - c)(1 - d)} + \sqrt {b(1 - c)(1 - d)(1 - a)} + \sqrt {c(1 - d)(1 - a)(1 - b)} + \sqrt {d(1 - a)(1 - b)(1 - c)} \le 1 + \sqrt {abcd}$ for $0\le a,b,c,d \le 1$
i thought that if the above inequality was true then i could conclude that the original inequality was only one case of the above inequality with $d = 0$ but then…. if i prove it with $'d'$ dependent $'a,b,c'$ then by assuming $d = 0$ i wud be assuming a relation for $a,b,c$ but that cant be done as the original inequality isnt homogenous.. so wat do i do?? i guess prove it immaterial wat $'d'$ we have.
and i have proved it for $a + b + c \ge 2$  but for the case $a + b + c < 2$ i am finding it difficullt to prove.. so i thought why not make the RHS larger as anyway if the value of d is immaterial then it should be so even in RHS so i replaced $\sqrt {abcd}$ by $\sqrt {abc}$ and this made me prove for $abc\ge\frac {1}{8}$ and i am stuck for the other case. The two cases i proved i made sole use of AM-GM directly..
i wonder if these will lead somewhere ❓