Cyclic Inequality Forms with Power 1/2,1/3 - Phạm Văn, Quốc (VNU Journal of Science: Mathematics - Physics; Vol 33 No 3DO - 10.25073/2588-1124/vnumap.4209, 2017)
The purpose of this paper is to establish inequalities between two terms \begin{equation*} F =\sum_{i=1}^{n}\sqrt{ax_{i}^{2}+bx_{i}x_{i+1}+cx_{i+1}^{2}+dx_i+ex_{i+1}+d }; \end{equation*} \begin{equation*} G =\sum_{i=1}^{n}\sqrt[3]{ax_{i}^{3}+bx_{i}^{2}x_{i+1}+cx_{i}x_{i+1}^{2}+dx_{i+1}^{3}}, \end{equation*} and $\sum_{i=1}^{n}x_{i}$ for a sequence of cyclic positive real numbers $ (x_{i})_{i=1}^{n+1}$ with $x_{n+1}=x_{1}$. The results depends on the sign of expressions containing the coefficients $a,b,c,d$. The general case for $F$ is also investigated.