  We continue our study on variability regions in
\cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region
of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z :
g({\mathbb D}) \subset \Omega, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots +
c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$
and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when
$\Omega\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of
the unit disk ${\mathbb D}$ onto $\Omega$. In the present article, we first
show that in the case $n=0$, $j=-1$ and $c=0$, the result obtained in
\cite{Ali-Vasudevarao-Yanagihara-2018} still holds when one assumes only that
$\Omega$ is starlike with respect to $P(0)$. Let $\mathcal{CV}(\Omega)$ be the
class of analytic functions $f$ in ${\mathbb D}$ with $f(0)=f'(0)-1=0$
satisfying $1+zf''(z)/f'(z) \in {\Omega}$. As applications we determine
variability regions of $\log f'(z_0)$ when $f$ ranges over
$\mathcal{CV}(\Omega)$ with or without the conditions $f''(0)= \lambda$ and
$f'''(0)= \mu$. Here $\lambda$ and $\mu$ are arbitrarily preassigned values. By
choosing particular $\Omega$, we obtain the precise variability regions of
$\log f'(z_0)$ for other well-known subclasses of analytic and univalent
functions.
