Generalization of an Inequality with Sum of Fractions

Abstract

Cvetkovski introduced the inequality: $\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}\geq a+b+c$ in his book \textit{Inequalities: Theorems, Techniques and Selected Problems}. Starting from there we generalize this inequality to the case with non-negative real number exponents: $\frac{a^m+b^m}{a^n+b^n}+\frac{b^m+c^m}{b^n+c^n}+\frac{c^m+a^m}{c^n+a^n}\geq a^{m-n}+b^{m-n}+c^{m-n}$ for $m\geq n$. Moreover, we also discuss the order of the sum of fractions when the exponents in the fractions change, and prove the inequality $\frac{a^m+b^m}{a^n+b^n}+\frac{b^m+c^m}{b^n+c^n}+\frac{c^m+a^m}{c^n+a^n}\geq\frac{a^{m-l}+b^{m-l}}{a^{n-l}+b^{n-l}}+\frac{b^{m-l}+c^{m-l}}{b^{n-l}+c^{n-l}}+\frac{c^{m-l}+a^{m-l}}{c^{n-l}+a^{n-l}}$ for real numbers $m\geq n\geq l\geq 0$. Versions of the two inequalities with $k$-variables are also examined.