Enveloping of the Values of an Analytic Function Related to the Number $e$

The problem of completely describing the approximation of the number $e$ by the elements of the sequence $(1+1/m)^m$, $m\in\mathbb{N}$, is considered. To this end, the function $f(z)=\exp\{(1/z)\ln(1+z)-1\}$, which is analytic in the complex plane with a cut along the half-line $(-\infty,-1]$ of the real line, is studied in detail. We prove that the power series $1+\sum^{\infty}_{n=1}(-1)^n a_n z^n$, where all $a_n$ are positive, which represents this function on the unit disk, envelops it in the open right half-plane. This gives a series of double inequalities for the deviation $e-(1+x)^{1/x}$ on the positive half-line, which are asymptotically sharp as $x\to 0$. Integral representations of the function $f(z)$ and of the coefficients $a_n$ are obtained. They play an important role in the study. A two-term asymptotics of the coeffients $a_n$ as $n\to \infty$ is found. We show that the coefficients form a logarithmically convex completely monotone sequence. We also obtain integral expressions for the derivatives of all orders of the function $f(z)$. It turns out that $f(x)$ is completely monotone on the half-line $x>-1$. Applications and development of the results are discussed.