  Very recently, the Fridman function of a complex manifold $X$ has been
identified as a dual of the squeezing function of $X$. In this paper, we prove
that the Fridman function for certain hyperbolic complex manifold $X$ is
bounded above by the injectivity radius function of $X$. This result also
suggests us to use the Fridman function to extend the definition of uniform
thickness to higher-dimensional hyperbolic complex manifolds. We also establish
an expression for the Fridman function (with respect to the Kobayashi metric)
when $X = \mathbb{D} \diagup \Gamma$ and $\Gamma$ is a torsion-free discrete
subgroup of isometries on the standard open unit disk $\mathbb{D}$. Hence,
explicit formulae of the Fridman functions for the annulus $A_r$ and the
punctured disk $\mathbb{D}^*$ are derived. These are the first explicit
non-constant Fridman functions. Finally, we explore the boundary behaviour of
the Fridman functions (with respect to the Kobayashi metric) and the squeezing
functions for regular type hyperbolic Riemann surfaces and planar domains
respectively.
