ratio functions

In this paper, we establish algebraic bounds of the ratio type in nature for the natural exponential function $e^x$ involving two parameters, a and n, which become optimal as a tends to 0 or n tends to infinity. The proof is mainly based on Chebyshev's integral inequality and properties of the incomplete gamma function. Subsequently, we focus on the simple case obtained with n = 1, with comparisons to existing literature results. For the applications, we provide alternative proofs of inequalities involving ratio functions of trigonometric and hyperbolic functions. Graphics are given to illustrate the theory.