Summary: This paper deals with the chemotaxis system with general sensitivity function: \[ \begin{cases} u_t =\nabla \cdot (\delta\nabla u-u\chi (v) \nabla v)\quad & x\in \Omega,\, t>0, \\ 0=\Delta v-v+u,\quad & x\in \Omega,\, t>0,\end{cases} \] under homogeneous Neumann boundary conditions in a bounded domain \(\Omega \subset\mathbb R^n\), \(n\geq 2\), with smooth boundary. Here, \(\delta >0\) and the initial function \(u(x,0)=u_0\) and the sensitivity function \(\chi\) satisfy: \[ u_0\in C^0(\bar{\Omega})\text{ with }\int_\Omega u_0 \mathrm{d}x>0, \]
\[ \chi (s)>0\text{ for } s>0\text{ and }\chi \in C^1([0,\infty)). \] We prove that the classical solutions to the above system are uniformly in-time-bounded provided that there exists a smooth positive function \(\varphi\) such that for some \(p>\frac{n}{2}\) and \(0<\lambda<1\) the following differential inequality holds \[ \varphi (s)+(p-1)\chi(s)\geq \sqrt{\frac{4\lambda\delta (p-1)}{p}\;\varphi'(s),}\quad s>0, \] where \(\varphi' (s)<0\) and \(s\varphi(s)\) is bounded from above. We also present our results for the special case \(0<\chi (s)\leq \frac{\chi_0}{s^k}\) with \(\chi_0>0\) and \(k\geq 1\). These results coincide with the results obtained by K. Fujie et al. [Math. Methods Appl. Sci. 38, No. 6, 1212–1224 (2015; Zbl 1329.35011)] in the case of \(k=1\) and extend their results in the case of \(k>1\).