Let \(M\) be an \(n\)-dimensional compact minimal submanifold in \(S^{n+p}\). \textit{A. M. Li} and \textit{J. M. Li} proved a scalar curvature pinching theorem [Arch. Math. 58, 582-594 (1992; Zbl 0767.53042)]. \textit{S.-T. Yau} proved a sectional curvature pinching theorem [Am. J. Math. 97, 76-100 (1975; Zbl 0304.53042)]. \textit{N. Ejiri} proved a Ricci curvature pinching theorem [J. Math. Soc. Japan 31, 251-256 (1979; Zbl 0396.53026)]. The present author has a try to generalize the above three results to compact pseudo-umbilical submanifolds in \(S^{n+p}\). He makes use of a method which is similar to \textit{B. Y. Chen}'s [Indiana Univ. Math. J. 20, 1175-1185 (1971; Zbl 0219.53047)]. It is unfortunate that Chen's paper contains a mistake [see Errata: Indiana Univ Math. J. 22, 399 (1972; Zbl 0244.53036)], which may imply that the proofs of theorems in the paper under review are not correct.