  For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient
function, a multiplicative arithmetic function defined by
\[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r),
& \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers
$\alpha$. The function $S_1$ is simply Euler's totient function $\phi$. Masser
and Shiu have established several fascinating results concerning sparsely
totient numbers, positive integers $n$ satisfying $\phi(n)<\phi(m)$ for all
integers $m>n$. We define a sparsely Schemmel totient number of order $r$ to be
a positive integer $n$ such that $S_r(n)>0$ and $S_r(n)<S_r(m)$ for all $m>n$
with $S_r(m)>0$. We then generalize some of the results of Masser and Shiu.
