A part-metric-related inequality chain and application to the stability analysis of difference equation. (English) Zbl 1133.26302

Summary: We find a new part-metric-related inequality of the form \(\min\{a_i,1/a_i:1\leq i\leq 5\}\leq ((1+w)a_1a_2a_3+a_4+a_5)/(a_1a_2+a_1a_3+a_2a_3+wa_4a_5)\leq \max{a_i,1/a_i:1\leq i\leq 5}\), where \(1\leq w \leq 2\). We then apply this result to show that \(\hat{c}=1\) is a globally asymptotically stable equilibrium of the rational difference equation \(x_n=(x_{n-1}+x_{n-2}+(1+w)x_{n-3}x_{n-4}x_{n-5})/(wx_{n-1}x_{n-2}+x_{n-3}x_{n-4}+x_{n-3}x_{n-5}+x_{n-4}x_{n-5})\), \(n=1,2,\dots, a_0,a_{-1},a_{-2},a_{-3},a_{-4}>0\).


26D05 Inequalities for trigonometric functions and polynomials
39A11 Stability of difference equations (MSC2000)
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