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 ❓

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