BdMO Online Forum

Prove that
\begin{eqnarray}
\left( x - 1 + \dfrac 1y \right) \left( y - 1 + \dfrac 1z \right)\leq \dfrac{x}{z}
\end{eqnarray}

$(x-1+\frac{1}{y})(y-1+\frac{1}{z})=xy-x+\frac{x}{z}-y+1-\frac{1}{z}+1-\frac{1}{y}+\frac{1}{yz}=\frac{x}{z}+2-(y+\frac{1}{y})\le \frac{x}{z}$ because $y+\frac{1}{y} \ge 2$
Similarly we get, $(y-1+\frac{1}{z})(z-1+\frac{1}{x}) \le \frac{y}{x}$ and $(z-1+\frac{1}{x})(x-1+\frac{1}{y}) \le \frac{z}{y}$ . Multiplying these three, we get the result.