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 satisfyingThen 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 thatThen for some , and

In this post, we consider the third result which can be stated as follows:

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