Let \(G\) be a compact group, and \(\widehat{G}\) be the set of equivalent classes of irreducible unitary representations of \(G\). One proves that, if \(f\) is a central integrable function such that \(f\) is \(L^ p\)- integrable in a neighborhood of the identity, and if \(\widehat {f}(\alpha) \geq 0\), \((\alpha \in \widehat{G})\), then \(\widehat {f}\in L^ q(\widehat {G})\), for \(q = p/p - 1\). One proves a similar result for a compact Gelfand pair. This theorem has been proved by Kawazoe and Miyazaki for compact semi-simple groups. In this note the author points out that one can drop the differentiability and semi-simplicity hypotheses.