Let \(L_{\infty}^{r}(\mathbb{R})\) be the space of functions \(f(x)\) which have locally absolutely continuous derivatives \(f^{(r-1)}(x)\) and a derivative \(f^{(r)}(x)\) such that \(\|f^{(r)}\|_{\infty}0\) and \(S_{\varepsilon}\) be some uniform net on \(\mathbb{R}\) with step \(\varepsilon.\) For \(f\in L_{\infty}(\mathbb{R})\) set \(\|f\|_{S_{\varepsilon}}= \sup_{x\in S_{\varepsilon}}|f(x)|.\) The authors obtain a strengthening of the Hörmander inequality for functions from \(L_{\infty}^{r}(\mathbb{R})\) in which instead of \(\|f\|_{\infty}\) the norm \(\|f\|_{S_{\varepsilon}}\) is used. Some assertions are proved. Among them the following one. If \(f\in L_{\infty}^{r}(\mathbb{R})\) is a \(2\pi\)-periodic function and \(\lambda\) is such that \[ \lambda^{-r}\varphi_{r}(\lambda\varepsilon/2)= {\|f\|_{S_{\varepsilon}}\over \|f^{(r)}\|_{\infty}}, \] where \(\varphi_{r}\) is a determined function, then for any \(k=0,1,\ldots,r-1\), the inequality \(\|f^{(k)}\|_{\infty}\leq \lambda^{-(r-k)} \|\varphi_{r-k}\|_{\infty} \|f^{(r)}\|_{\infty}\) holds true. A function from \(L_{\infty}^{r}\) exists such that the last inequality becomes an equality.