a,b,c are positive real numbers and abc =1 prove that (a−1+(1/b))(b−1+(1/c))(c−1+(1/a))≤1


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Question Number 195790 by justenspi last updated on 10/Aug/23

$a, b, c a r e p o s i t i v e r e a l n u m b e r s a n d a b c = 1$ $p r o v e t h a t$ $(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a}) ⩽ 1$
Commented by justenspi last updated on 10/Aug/23

$C a s e (I) (a - 1 + \frac{1}{b}), (b - 1 + \frac{1}{c}), (c - 1 + \frac{1}{a}) > 0$ $\Rightarrow \sqrt[3]{(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})} ⩽ \frac{a + b + c + a b + b c + a c - 3}{3} . . . . . . (i)$ $\Rightarrow \sqrt[3]{(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})} ⩽ \frac{a b + b c + a c - a - b - c + 3}{3} . . . . . . (i i)$ $\Rightarrow 2 \sqrt[3]{(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})} ⩽ \frac{2 (a b + b c + a c)}{3}$ $\Rightarrow \sqrt[3]{(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})} ⩽ \frac{a b + b c + a c}{3}$ $t h e e q u a l i t y h o l d s w h e n$ $a + b + c = a b + b c + a c \land a b c = 1 \land a, b, c > 0$ $\Rightarrow a + b + c = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \Rightarrow a = 1, b = 1, c = 1$ $h e n c e t h e m a x i m u m v a l u e o f$ $\sqrt[3]{(a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})} \to \frac{1 + 1 + 1}{3} = 1$ $\Rightarrow (a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a}) m a x = 1$ $C a s e (I I) A t m o s t o n e t h e f a c t o r s i s n e g a t i v e$ $h e n c e w e a r e d o n e$
Commented by justenspi last updated on 10/Aug/23

$i s t h a t r i g h t, p l e a s e h e l p$
	Answered by York12 last updated on 10/Aug/23

	$s e t a = \frac{x}{y}, b = \frac{y}{z}, c = \frac{z}{x}$ $\Rightarrow (a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a})$ $= \frac{(x - y + z) (x + y - z) (- x + y + z)}{x y z}$ $\sqrt{(x - y + z) (x + y - z)} ⩽ \frac{x - y + z + x + y - z}{2} = x . . . . (i)$ $\sqrt{(x + y - z) (- x + y + z)} ⩽ \frac{x + y - z - x + y + z}{2} = y . . . . (i i)$ $\sqrt{(x - y + z) (- x + y + z)} ⩽ \frac{x - y + z - x + y + z}{2} = z . . . . (i i i)$ $\Rightarrow (i) \times (i i) \times (i i i) \Rightarrow (x - y + z) (x + y - z) (- x + y + z) ⩽ x y z$ $\Rightarrow \frac{(x - y + z) (x + y - z) (- x + y + z)}{x y z} ⩽ 1$ $\Rightarrow (a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a}) ⩽ 1$ $I n c a s e o n e o f t h e f a c t o r s i s n e g a t i v e$ $\Rightarrow (a - 1 + \frac{1}{b}) (b - 1 + \frac{1}{c}) (c - 1 + \frac{1}{a}) ⩽ 0 < 1$
	Commented by justenspi last updated on 10/Aug/23

	$t h a n k s b u t i s m y s o l u t i o n r i g h t$
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