A Part-Metric-Related Inequality Chain and Application to the Stability Analysis of Difference Equation

We find a new part-metric-related inequality of the form min{a i,1/ai:1i5} ((1+w)a1a2a3+a4+ a5)/(a1a2+a1a3+a2a3+wa4a5) max{a i,1/ai:1i5}, where 1 w 2. We then apply this result to show that c=1 is a globally asymptotically stable equilibrium of the rational difference equation xn=(xn-1+xn-2+(1+w)xn-3xn-4 xn-5)/ (wx n-1xn-2+xn-3xn-4+xn-3xn&#x 2212;5+xn -4xn-5) , n =1,2,..., a0,a -1,a-2,a&#x 2212;3,a-4 > 0.