Generalizing Inequalities Using Power Series Approach

Abstract

In 1903 Nesbitt introduced a famous inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}$ for any positive real numbers $a$, $b$ and $c$. Among all its proofs, Mortici provided a unique approach applying the convergence of power series together with the power means inequality. Adopting this technique, we first generalize several Nesbitt type inequalities to $n$ variable versions. We then combine the knowledge of power series, Young's inequality, and the rearrangement inequality, and deduce some new inequalities.