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Question Number 150887 by mathdanisur last updated on 16/Aug/21

Commented by dumitrel last updated on 16/Aug/21

$a ⩽ b ⩽ c ⩽ d \Rightarrow {(a + b + c + d)}^{2} ⩾ 8 (a c + b d)$
Commented by mathdanisur last updated on 16/Aug/21

$How Ser, x; y or a; b$
Commented by dumitrel last updated on 16/Aug/21

$a = m_{h}; b = m_{g}; c = m_{a}; d = m_{p}$
Commented by mathdanisur last updated on 16/Aug/21

$Dear Ser, l did not understand$
	Answered by dumitrel last updated on 16/Aug/21

	$\Leftrightarrow {(\frac{m_{h}}{2} + \frac{m_{g}}{2} + \frac{m_{a}}{2} + \frac{m_{p}}{2})}^{2} ⩾ 2 \cdot \frac{2 x y}{x + y} \cdot \frac{x + y}{2} + 2 \sqrt{x y} \cdot \sqrt{\frac{x^{2} + y^{2}}{2}} \Leftrightarrow$ ${(m_{h} + m_{g} + m_{a} + m_{p})}^{2} ⩾ 8 (m_{h} m_{a} + m_{g} m_{p})$ $f (x) = 2 x^{2} + x (m_{h} + m_{g} + m_{a} + m_{p}) + m_{h} m_{a} + m_{g} m_{p}$ $\Rightarrow f (x) = (x + m_{h}) (x + m_{a}) + (x + m_{g}) (x + m_{p})$ $f (m_{h}) = (- m_{h} + m_{g}) (- m_{h} + m_{p}) ⩾ 0$ $f (- m_{a}) = (- m_{a} + m_{g}) (- m_{a} + m_{p}) ⩽ 0 \Rightarrow △_{f} ⩾ 0 \Rightarrow$ ${(m_{h} + m_{g} + m_{a} + m_{p})}^{2} ⩾ 8 (m_{h} m_{a} + m_{g} m_{p})$
	Commented by mathdanisur last updated on 16/Aug/21

	$Cool S er Thank You$
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