  We derive an improved Poincar\'e inequality in connection with the
Babu\v{s}ka-Aziz and Friedrichs-Velte inequalities for differential forms by
estimating the domain specific optimal constants figuring in the respective
inequalities with each other provided the domain supports the Hardy inequality.
We also derive upper estimates for the constants of a star-shaped domain by
generalizing the known Horgan-Payne type estimates for planar and spatial
domains to higher dimensional ones.
