Densely algebraic bounds for the exponential function

An upper bound for ex that implies the inequality between the arithmetic and geometric means is generalized with the introduction of a new parameter n. The new upper bound is smoothly and densely algebraic in n, and valid for -b < x < 1 for arbitrarily large positive b provided that n (> 1) is sufficiently close to 1. The range of its validity for negative x is investigated through the study of a certain family of quadrinomials.