Complexity Analysis of the Rule based Cauchy’s Inequality in solving the n-order Differential Equations

This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ ex where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities.