Estimates for the hyperbolic and quasihyperbolic metrics in hyperbolic regions

In this paper we consider ordinary derivative of universal covering mappings $f$ of hyperbolic regions $D$ in the complex plane. We obtain sharp bounds for the ratio $\mathrm{\mid }{f}^{\mathrm{\prime }}(z)\mathrm{\mid }\mathrm{/}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(f(z),\mathrm{\partial }f(D))$ in terms of the hyperbolic density in simply connection domains. In arbitrary domains, we find a necessary and sufficient condition for an upper bound for the quantity $|f’(z)|/{\rm dist}(f(z),\partial f(D))$ to hold in terms of the hyperbolic density. As an application of the above results, it is observed that the bounds for the quantity of the above type are closely connected with similar bounds for $\mathrm{\mid }{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)\mathrm{\mid }$.