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We find a new part-metric-related inequality of the form min{ai,1/ai:1Ã¢Â‰Â¤iÃ¢Â‰Â¤5}Ã¢Â‰Â¤((1+w)a1a2a3+a4+a5)/(a1a2+a1a3+a2a3+wa4a5)Ã¢Â‰Â¤max{ai,1/ai:1Ã¢Â‰Â¤iÃ¢Â‰Â¤5}, where 1Ã¢Â‰Â¤wÃ¢Â‰Â¤2. We then apply this result to show that  c^=1  is a globally asymptotically stable equilibrium of the rational difference equation xn=(xnÃ¢ÂˆÂ’1+xnÃ¢ÂˆÂ’2+(1+w)xnÃ¢ÂˆÂ’3xnÃ¢ÂˆÂ’4xnÃ¢ÂˆÂ’5)/(wxnÃ¢ÂˆÂ’1xnÃ¢ÂˆÂ’2+xnÃ¢ÂˆÂ’3xnÃ¢ÂˆÂ’4+xnÃ¢ÂˆÂ’3xnÃ¢ÂˆÂ’5+xnÃ¢ÂˆÂ’4xnÃ¢ÂˆÂ’5), n=1,2,Ã¢Â€Â¦,a0,aÃ¢ÂˆÂ’1,aÃ¢ÂˆÂ’2,aÃ¢ÂˆÂ’3,aÃ¢ÂˆÂ’4>0

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