For given k \in (0,1) and r > 0 , a self-mapping T: \, M \to M is said to be r -roughly k -contractive provided \|Tx - Ty\| \le k\,\|x - y\| + r \quad (x,y \in M). To state fixed-point properties of such a mapping, the self-Jung constant J_s(X) is used, which is defined as the supremum of the ratio 2\,r_{\mathrm{conv} S}(S)/\mathrm{diam} S over all non-empty, non-singleton and bounded subsets S of some normed linear space X , where r_{\mathrm{conv} S}(S) = \inf_{x\in\mathrm{conv} S} \sup_{y\in S} \|x - y\| is the self-radius of S and \mathrm{diam} S is its diameter. If M is a closed and convex subset of some finite-dimensional normed space X and if T: \, M \to M is r -roughly k -contractive, then for all \varepsilon > 0 there exists x^* \in M such that \|x^* - Tx^*\| < \frac12\,J_s(X)\, r + \varepsilon. If \mathrm{dim} X = 1 , or X is some two-dimensional strictly convex normed space, or X is some Euclidean space, then there is x^* \in M satisfying \|x^* - Tx^*\| \le {1 \over 2}\,J_s(X)\,r .