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Determine the greatest real number

Source: Austrian Mathematical Olympiad 2018

June 12, 2018
inequalitiesAustriaAUT

Problem Statement

Let α\alpha be an arbitrary positive real number. Determine for this number α\alpha the greatest real number CC such that the inequality(1+αx2)(1+αy2)(1+αz2)≥C(xz+zx+2)\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) is valid for all positive real numbers x,yx, y and zz satisfying xy+yz+zx=α.xy + yz + zx =\alpha. When does equality occur? (Proposed by Walther Janous)
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