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Question Number 155594 by mathdanisur last updated on 02/Oct/21

	Answered by ghimisi last updated on 02/Oct/21

	$\Leftrightarrow Σ (\sqrt{x^{4} + y^{4}} - x^{2} - y^{2} + (2 - \sqrt{2}) x y) ⩾ 0$ $\sqrt{x^{4} + y^{4}} - x^{2} - y^{2} + (2 - \sqrt{2}) x y ⩾ 0 \Leftrightarrow$ $(2 - \sqrt{2}) x y ⩾ x^{2} + y^{2} - \sqrt{x^{4} + y^{4}} \Leftrightarrow$ $(2 - \sqrt{2}) x y ⩾ \frac{2 x^{2} y^{2}}{x^{2} + y^{2} + \sqrt{x^{4} + y^{4}}} \Leftrightarrow x^{2} + y^{2} + \sqrt{x^{4} + y^{4}} ⩾ (2 + \sqrt{2}) x y (∙)$ $x^{2} + y^{2} ⩾ 2 x y (1)$ $\sqrt{x^{4} + y^{4}} ⩾ \sqrt{2 x^{2} y^{2}} = x y \sqrt{2} (2)$ $(1) + (2) \Rightarrow (∙)$
	Commented by mathdanisur last updated on 02/Oct/21

	$Perfect dear S er, thank you so much$
	Commented by amin96 last updated on 02/Oct/21

	$g r e a t$
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