Let \(M^n\) be a compact space-like hypersurface in de Sitter space and let \(H_k\) denotes \(k\)-th mean curvature function of \(M^n\). The authors prove that if \(M^n\) is contained in the chronological future (or past) of an equator of de Sitter space then \(M^n\) is a totally umbilical round sphere if there exist nonnegative constants \(C_1,C_2,\dots,C_{l-1}\), at least one \(C_i\) is positive, such that \(H_l=\sum_{i=1}^{l-1}C_iH_i\,\). This result extends the previous result of \textit{L. J. Alias} and \textit{S.-E. Koh} [J. Geom. Phys. 39, 45--49 (2001; Zbl 0988.53007)].