Bounds for the Wallis ratio]{Bounds for the Wallis ratio with two parameters

Abstract

In this paper, we establish several new bounds for the Wallis ratio $%
W_{n}=\left( 2n-1\right) !!/\left( 2n\right) !!$ in the form of $1/\sqrt{\pi
G_{n}\left( a,b\right) }$, where%
\begin{equation*}
G_{n}\left( a,b\right) =n+\frac{1}{4}+\frac{b}{n+a}
\end{equation*}%
with $a>-1$ and $b>-5\left( a+1\right) /4$. In particular, we find that%
\begin{equation*}
\frac{1}{\sqrt{\pi G_{n}\left( 1/4,1/32\right) }}<W_{n}<\frac{1}{\sqrt{\pi
G_{n}\left( a_{0},1/32\right) }}
\end{equation*}%
for $n\in \mathbb{N}$ with the best constants $1/4$ and $a_{0}=5\left(
16-5\pi \right) /\left( 16\pi \right) $, where the lower and upper bounds
are the sharpest.