An asymptotic estimate of Aoki’s function

Abstract

The Aoki's function \(A(x):=\left(1+\frac{1}{x}\right)^x+\left(1-\frac{1}{x}\right)^{-x}\) is sharply estimated for \(x\gg1\). For example, we have the zero approximation given as
\[
2e\left(1+\frac{1}{4x^2-1}\right)<A(x)
<2e\left(1+\frac{3}{4x^2-1}\right),\quad x\ge \frac{29}{14}.
\]