An Alternative Proof for an Inequality with Cyclic Fractions

Abstract

It is known that $\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b}$ for any positive $a, b, c$. In this paper we provide an alternative proof for this inequality, and generalize it to an $n$-variable version.