Optimal bounds for a Lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

In this paper the interpolation inequality of Szepessy [12, Lemma 4.2] is revisited. The lower bound in the above reference is proven to be proportional to p−2, where p is a polynomial degree, that goes fast to zero as p increases. We prove that the lower bound is proportional to ln2⁡p which is an increasing function. Moreover, we prove that this estimate is sharp.