Simultaneously, I have recently found the following identity in the same fashion of the Picone identity for . It says that

for any function . It is worth noticing that the original Picone identity says that

for any function . It turns out that a few days ago, this identity appeared in a recent notes by Dwivedi and Tyagi, see Lemma 2.1 from here. The extra term

naturally appears since it only involves up to third order derivatives. However, to compare this term with , we only need to assume that has a fixed sign. To see how this identity could be useful, let us consider the following equation

naturally arises from prescribing -curvature in Riemannian manifolds of dimension .