In the article, we present several sharp bounds for the modified Bessel function of the first kind
$I_{0}(t)=\sum_{n=0}^{\infty}\frac{t^{2n}}{2^{2n}(n!)^{2}}$
and the Toader-Qi mean
$TQ(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}a^{\cos^{2}\theta }b^{\sin^{2}\theta}\,d\theta$
for all
$t>0$
and
$a, b>0$
with
$a\neq b$
.
