lean_workbook_plus_13220

Prove that for positive real numbers x, y, and z, the following inequality holds:\n\n(1) $\frac{z+x}{y}+\frac{x+y}{z} \geq \frac{4x}{y+z}+\frac{8yz}{(y+z)^2}$\n\n(2) $\frac{z+x}{y}+\frac{x+y}{z} \geq \frac{4x}{y+z}+\frac{(y+z)^2}{y^2+z^2}$\n\nIt is given that the second inequality is stronger than the first and can be proven using Cauchy-Schwarz (CS) and Arithmetic-Geometric Mean (AG) inequalities.