  We consider certain inequalities among the Apollonian metric, the Apollonian
inner metric, the $j$ metric and the quasihyperbolic metric. We verify that
whether these inequalities can occur in simply connected planar domains and in
proper subdomains of $\mathbb{R}^n~(n\ge 2)$. We have seen from our
verification that most of the cases cannot occur. This means that there are
many restrictions on domains in which these inequalities can occur. We also
consider two metrics $j$ and $d$, and investigate whether a plane domain
$D\varsubsetneq\mathbb{C}$, for which there exists a constant $c>0$ with
$j(z,w) \le c\, d(z,w)$ for all $z,w \in D$, is a uniform domain. In
particular, we study the case when $d$ is the $\lambda$-Apollonian metric. We
also investigate the question, whether simply connected quasi-isotropic domains
are John disks and conversely. Isometries of the quasihyperbolic metric, the
Ferrand metric and the K--P metric are also obtained in several specific
domains in the complex plane.
  In addition to the above, some problems on univalent functions theory are
also solved. We denote by $\mathcal{S}$, the class of normalized univalent
analytic functions defined in the unit disk. We consider some geometrically
motivated subclasses, say $\mathcal{F}$, of $\mathcal{S}$. We obtain the
largest disk $|z|<r$ for which $\frac{1}{r}f(rz)\in \mathcal{F}$ whenever $f\in
\mathcal{S}$. We also obtain necessary and sufficient coefficient conditions
for $f$ to be in $\mathcal{F}$. Finally, we present the pre-Schwarzian norm
estimates of functions from $\mathcal{F}$ and that of certain convolution or
integral transforms of functions from $\mathcal{F}$.
