8.3.4 Order-optimality of the factor \(n+1\)
For all \(n\),
\[ \frac{n+1}{2}\, h_n(I) \; \le \; c_n(I) \; \le \; a_n(I) , \]
whereas the maximal difference theorem gives \(a_n \le e\, (n+1)\, h_n\) in general. Hence no bound \(a_n \le \varphi (n)\, h_n\) can hold with \(\varphi (n) = o(n)\).