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

On the inequalities of Babu\v{s}ka-Aziz, Friedrichs and Horgan-Payne