Let a,b,c ∈R^+ , a+b+c=3 prove the following inequality (((2a−3)^2 )/b)+(((2b−3)^2 )/c)+(((2c−3)^2 )/a)≥((a^2 +b^2 )/(a+b))+((b^2 +c^2 )/(b+c))+((c^2 +a^2 )/(c+a))


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Question Number 202348 by York12 last updated on 24/Dec/23

$Let a, b, c \in R^{+}, a + b + c = 3 prove the following inequality$ $\frac{{(2 a - 3)}^{2}}{b} + \frac{{(2 b - 3)}^{2}}{c} + \frac{{(2 c - 3)}^{2}}{a} ⩾ \frac{a^{2} + b^{2}}{a + b} + \frac{b^{2} + c^{2}}{b + c} + \frac{c^{2} + a^{2}}{c + a}$
	Answered by mrdiane last updated on 25/Dec/23
	$on a 0(a+b+c)^2 =a^2 +b^2 +2(ab+ac+bc comme a,b et >0donc (a+b+c)^2 ≥a^2 +b^2 or b≤a+b donc on a { (((a+b+c)^2 ≥a^2 +b^2 (1))),(((1/b)≥(1/(a+b)) (2))) :} (1)×(2) donne (((a+b+c)^2 )/b)≥ ((a^2 +b^2 )/(a+b))⇔(((a+a−3)^2 )/b)≥((a^2 +b^2 )/(a+b)) pae annalogie on a { (((((2a−3)^2 )/b)≥((a^2 +b^2 )/(a+b)) (3))),(((((2b−3)^2 )/c)≥((b^2 +c^2 )/(b+c)) (4))),(((((2c−3)^2 )/a)≥((c^2 +a^2 )/(a+c)) (5))) :} (3)+(4)+(5) donne (((2a−3)^2 )/b)+(((2b−3))/c)+(((2c−3))/a)≥((a^2 +b^2 )/(a+b))+((b^2 +c^2 )/(b+c))+((c^2 +a^2 )/(c+a))$
	$o n a 0 {(a + b + c)}^{2} = a^{2} + b^{2} + 2 (a b + a c + b c c o m m e a, b e t > 0 d o n c$ ${(a + b + c)}^{2} ⩾ a^{2} + b^{2} o r b ⩽ a + b$ $d o n c o n a {\begin{cases} {(a + b + c)}^{2} ⩾ a^{2} + b^{2} (1) \\ \frac{1}{b} ⩾ \frac{1}{a + b} (2) \end{cases}$ $(1) \times (2) d o n n e \frac{{(a + b + c)}^{2}}{b} ⩾ \frac{a^{2} + b^{2}}{a + b} \Leftrightarrow \frac{{(a + a - 3)}^{2}}{b} ⩾ \frac{a^{2} + b^{2}}{a + b}$ $p a e a n n a l o g i e o n a {\begin{cases} \frac{{(2 a - 3)}^{2}}{b} ⩾ \frac{a^{2} + b^{2}}{a + b} (3) \\ \frac{{(2 b - 3)}^{2}}{c} ⩾ \frac{b^{2} + c^{2}}{b + c} (4) \\ \frac{{(2 c - 3)}^{2}}{a} ⩾ \frac{c^{2} + a^{2}}{a + c} (5) \end{cases}$ $(3) + (4) + (5) d o n n e \frac{{(2 a - 3)}^{2}}{b} + \frac{(2 b - 3)}{c} + \frac{(2 c - 3)}{a} ⩾ \frac{a^{2} + b^{2}}{a + b} + \frac{b^{2} + c^{2}}{b + c} + \frac{c^{2} + a^{2}}{c + a}$
	Commented by York12 last updated on 25/Dec/23

	Merci monsieur, mais je ne comprends pas la première ligne
	Commented by York12 last updated on 25/Dec/23

	${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c)$ $\neq a^{2} + b^{2} + 2 (a b + b c + a c)$ $b ⩾ b ⇎ b ⩾ a + b \to (because eauality holds when a = 0 b u t a > 0 \Rightarrow The equality does not hold)$ ${(a + b + c)}^{2} ⩾ a^{2} + b^{2} \to (I do not see the equality case)$ $\frac{{(a + b + c)}^{2}}{b} ⩾ \frac{a^{2} + b^{2}}{a + b} \Leftrightarrow \frac{{(a + a - 3)}^{2}}{b} ⩾ \frac{a^{2} + b^{2}}{a + b} \to (Why)$
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