For a normalized analytic function
$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$
in the unit disk
$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$
, the estimate of the integral means
$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$
is an important quantity for certain problems in fluid dynamics, especially when the functions
$f(z)$
are nonvanishing in the punctured unit disk
$\mathbb{D}\setminus \{0\}$
. Let
${\rm\Delta}(r,f)$
denote the area of the image of the subdisk
$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$
under
$f$
, where
$0<r\leq 1$
. In this paper, we solve two extremal problems of finding the maximum value of
$L_{1}(r,f)$
and
${\rm\Delta}(r,z/f)$
as a function of
$r$
when
$f$
belongs to the class of
$m$
-fold symmetric starlike functions of complex order defined by a subordination relation. One of the particular cases of the latter problem includes the solution to a conjecture of Yamashita, which was settled recently by Obradović
et al. [‘A proof of Yamashita’s conjecture on area integral’,
Comput. Methods Funct. Theory13 (2013), 479–492].