Densely algebraic bounds for the exponential function

An upper bound for $e^{x}$ 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.