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|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}

Source: Germany 1999 p5

February 23, 2020
inequalitiesalgebra

Problem Statement

Consider the following inequality for real numbers x,y,zx,y,z: ∣x−y∣+∣y−z∣+∣z−x∣≤ax2+y2+z2|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2} . (a) Prove that the inequality is valid for a=22a = 2\sqrt2 (b) Assuming that x,y,zx,y,z are nonnegative, show that the inequality is also valid for a=2a = 2.
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