We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let \documentclass[12pt]{minimal}
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\begin{document}$$M^n$$\end{document} be a compact hypersurface with constant mean curvature \documentclass[12pt]{minimal}
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\begin{document}$$H$$\end{document} in \documentclass[12pt]{minimal}
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\begin{document}$$S^{n+1}$$\end{document}. Denote by \documentclass[12pt]{minimal}
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\begin{document}$$S$$\end{document} the squared norm of the second fundamental form of \documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document}. We prove that there exist two positive constants \documentclass[12pt]{minimal}
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\begin{document}$$\gamma (n)$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\delta (n)$$\end{document} depending only on \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} such that if \documentclass[12pt]{minimal}
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\begin{document}$$|H|\le \gamma (n)$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\beta (n,H)\le S\le \beta (n,H)+\delta (n)$$\end{document}, then \documentclass[12pt]{minimal}
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\begin{document}$$S\equiv \beta (n,H)$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document} is one of the following cases: (i) \documentclass[12pt]{minimal}
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\begin{document}$$S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$$\,1\le k\le n-1$$\end{document}; (ii) \documentclass[12pt]{minimal}
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\begin{document}$$S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$$\end{document}. Here \documentclass[12pt]{minimal}
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\begin{document}$$\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$$\end{document}.
