On connections between domain specific constants in some norm inequalities

We derive connections between optimal domain specific constants figuring in the Friedrichs-Velte inequality for conjugate harmonic functions, in the Babu\v{s}ka-Aziz inequality for the divergence and in the improved Poincar'e inequality for the gradient. With the same method we obtain for spatial domains an improved Poincar'e inequality for the rotation in connection with the corresponding Babu\v{s}ka-Aziz inequality.