A domain \(\Omega\subset {\mathbb C}^n\) is called \({\mathbb C}\)-convex, if its intersection with any complex line is contractible. For smooth bounded \(\Omega\) this condition is equivalent to the condition that for any boundary point \(\zeta \) of \(\Omega\) the holomorphic tangent space does not trespass \(\Omega\). By \(d_\Omega^B\), \(d_\Omega^C\) and \(d_\Omega^K\) let us denote the invariant pseudodistance functions of Bergman, Carathéodory, and Kobayashi, respectively.
Then the author establishes lower bounds for the above distances in terms of the boundary distance \(\delta_\Omega\) of \(\Omega\), more precisely, his results are:
(1) For any proper \({\mathbb C}\)-convex domain \(D \subset {\mathbb C}^n\) one has \[ d_D^C (z,w) \geq \frac{1}{4}\log \,\frac{\delta_D(z)}{\delta_D(w)}. \]
(2) If \(D\) is bounded, then, for any compact \(K \subset D\) there is a constant \(c_K>0\) such that \[ d_D^B(z,w)\geq \frac{1}{4} \log \,\frac{1}{\delta_D(w)} - c_K \] for any \(z \in K\), \(w \in D\).
These estimates apply also to the Kobayashi distance, since \(d_D^K , d_D^B\geq d_D^C\) and one has also, by work of Nikolov-Pflug-Zwonek, \(d_D^K \leq 4 d_D^B \leq c_n d_D^K\) with some constant \(c_n\), for any \({\mathbb C}\)-convex domain \(D \subset {\mathbb C}^n\).
The author also obtains analogous results in the special case of convexifiable domains in \({\mathbb C}^n\). Finally he proves a sharp lower and upper estimate for \(d_D^C\) on a planar domain with a Dini-smooth boundary.