  The Toader-Qi mean of positive numbers $a$ and $b$ defined by
\begin{equation*} TQ\left( a,b\right) =\frac{2}{\pi }\int_{0}^{\pi /2}a^{\cos
^{2}\theta }b^{\sin ^{2}\theta }d\theta \end{equation*} is related to the
modified Bessel function of the first kind. In this paper, we present several
properties of this mean, and establish some sharp inequalities for this mean in
terms of power and logarithmic means. From these a nice chain of inequalities
involving Gauss compound mean, Toader mean and Toader-Qi mean is presented.
