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].