  The equivalence between the inequalities of Babu\v{s}ka-Aziz and Friedrichs
for sufficiently smooth bounded domains in the plane has been shown by Horgan
and Payne 30 years ago. We prove that this equivalence, and the equality
between the associated constants, is true without any regularity condition on
the domain. For the Horgan-Payne inequality, which is an upper bound of the
Friedrichs constant for plane star-shaped domains in terms of a geometric
quantity known as the Horgan-Payne angle, we show that it is true for some
classes of domains, but not for all bounded star-shaped domains. We prove a
weaker inequality that is true in all cases.
