Bounding Taylor approximation errors for the exponential function in the presence of a power weight function

Abstract

Motivated by the needs in the theory of large deviations and in the theory of Lundberg's equation with heavy-tailed distribution functions, we study for $n=0,1,...$ the maximization of $S:(1-e^{-s}(1+\frac{s^1}{1!}+...+\frac{s^n}{n!}\left)\right)/s^{\delta }=E_{n,\delta }(s)$ over $s\ge 0$, with $\delta \in (0,n+1)$, $U:(-1{)}^{n+1}(e^{-u}-(1-\frac{u^1}{1!}+...+(-1{)}^n\frac{u^n}{n!}\left)\right)/u^{\delta }=G_{n,\delta }(u)$ over $u\ge 0$ with $\delta \in (n,n+1)$. We show that $E_{n,\delta }(s)$ and $G_{n,\delta }(u)$ have a unique maximizer $s=s_n(\delta )>0$ and $u=u_n(\delta )>0$ that decrease strictly from $+\infty$ at $\delta =0$ and $\delta =n$, respectively, to 0 at $\delta =n+1$. We use Taylor's formula for truncated series with remainder in integral form to develop a criterion to decide whether a particular smooth function $S(\delta )$, $\delta \in (0,n+1)$, or $U(\delta )$, $\delta \in (n,n+1)$, respectively, is a lower/upper bound for $s_n(\delta )$ and $u_n(\delta )$, respectively. This criterion allows us to find lower and upper bounds for $s_n$ and $u_n$ that are reasonably tight and simple at the same time. Furthermore, as a consequence of the identities $\frac{d}{d\delta }[\ln M{E}_{n,\delta }]=-\ln s_n(\delta )$ and $\frac{d}{d\delta }[\ln M{G}_{n,\delta }]=-\ln u_n(\delta )$, we show that $M{E}_{n,\delta }$ and $M{G}_{n,\delta }$ are log-convex functions of $\delta \in (0,n+1)$ and $\delta \in (n+1,n)$, respectively, with limiting values 1 ($\delta \downarrow 0$) and $1/(n+1)!$ ($\delta \uparrow n+1$) for $E$, and $1/n!(\delta \downarrow n)$ and $1/(n+1)!(\delta \uparrow n+1)$ for $G$. The minimal values ${\hat{E}}_n$ and ${\hat{G}}_n$ of $M{E}_{n,\delta }$ and $M{G}_{n,\delta }$, respectively, as a function of $\delta$, as well as the minimum locations ${\delta }_{n,E}$ and ${\delta }_{n,G}$ are determined in closed form.