Cyclic Inequality Forms with Power 1/2,1/3

Abstract

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.