MathDB

Let

\alpha

be an arbitrary positive real number. Determine for this number

\alpha

the greatest real number

C

such that the inequality

\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, y

and

z

satisfying

xy + yz + zx =\alpha.

When does equality occur? (Proposed by Walther Janous)

1