The author proves the following inequality of Hardy type, which is valid for \(u \in C_0^\infty(\mathbb R^n_+)\): \[ \int_{\mathbb R^n_+} |\nabla u|^2 \,dx \geq {1\over 4} \int_{\mathbb R^n_+} {|u|^2 \over x_n^2} \,dx + {1\over 8} \int_{\mathbb R^n_+} {|u|^2 \over x_n(x_{n-1}^2 + x_n^2)} \,dx. \tag{1} \] This is an improvement of a previous result by V. G. Maz'ja, who showed a similar estimate, with \({1\over 16}\) replacing \({1\over 8}\). Next, the following inequality is proved, solving in affermative a conjecture, again formulated by Maz'ja: if \(u \in C_0^\infty(\mathbb R^n_+)\), \(p \in (1, +\infty)\), \(\tau \in (0, 1]\), \[ \int_{\mathbb R^n_+} |\nabla u|^p \,dx \geq {p-1\over p} \int_{\mathbb R^n_+} {|u|^p \over x_n^p} \,dx + \alpha(p,\tau) \int_{\mathbb R^n_+} {|u|^p \over x_n^{p-\tau}(x_{n-1}^2 + x_n^2)^{\tau \over 2}} \,dx, \tag{2} \] with \(\alpha(p,\tau)\) depending only on \(p\) and \(\tau\).