In this post, we proved the following result (appeared in a paper by Y.Y. Li published in J. Eur. Math. Soc. (2004))
Lemma 1. For and , let be a function defined on and valued in satisfying
Then is constant or .
Later, we considered the equality case in this post and proved the following result:
Lemma 2. Let , and . Suppose that for every there exists such that
Then for some , and
In this post, we consider the third result which can be stated as follows: