We give the maximal order of a class of
multiplicative
arithmetical functions, including certain functions of the type
![$\sigma A(n)=
\sum {d \in A(n)} d$](img1.gif)
, where
![$A(n)$](img2.gif)
is a subset of the set of all
positive
divisors of
![$n$](img3.gif)
. As special cases we obtain the maximal orders of the
divisor-sum function
![$ \sigma(n)$](img4.gif)
and its analogues
![$ \sigma^*(n)$](img5.gif)
,
![$ \sigma^{(e)} (n)$](img6.gif)
,
![$ \sigma^{**} (n)$](img7.gif)
,
![$ \sigma A(n)$](img8.gif)
, representing the
sum of
unitary divisors, exponential divisors, bi-unitary divisors and
elements
of a regular system
![$A$](img9.gif)
of divisors of
![$n$](img3.gif)
, respectively.
We also give the minimal order of another class of
multiplicative functions, including the Euler function
![$ \phi(n)$](img10.gif)
, its
unitary
analogue
![$\phi^*(n)$](img11.gif)
and their common generalizations.
We pose the problem of finding the maximal order of a
![$ \sigma$](img12.gif)
-type
function
not covered by our results.