A new improvement of Jensen's inequality
Let \(f:C \to\mathbb{R}\) be a convex mapping on a convex subset \(C\) of a linear space. For \(x_i\in C\), \(p_i \geq 0\), \(1\leq i\leq n\), \(P_n: =\sum^n_{i=1} p_i>0\), \(1\leq k<n\), and \(0 \leq t\leq 1\) define \[ L_n^{[k+1]} (t): =P_n^{-k-1} \sum^n_{i_1, \dots, i_{k+1} = 1} p_{i_1} \dots p_{i_{k+1}} f \bigl(t(x_{i_1} + \cdots + x_{i_k})/k + (1-t) x_{i_{k+1}} \bigr). \] Then the author proves the following main result. On \(0\leq t\leq 1\quad L_n^{[k+1]} (t)\) is convex and satisfies \[ L_n^{[k+1]} \bigl(k/(k+1) \bigr) \leq L_n^{[k+1]} (t)\leq t L_n^{[k]} \bigl((k-1)/k \bigr)+ (1-t) L_n^{[1]} (0)\leq L_n^{[k+1]} (0)= L_n^{[1]} (0). \] Furthermore, \(L_n^{[k+1]} \searrow\) on \([0,k/(k+1)]\) and \(\nearrow\) on \([k/(k+1),1]\). As application numerous new inequalities are derived for concave \(\log f\), for sublinear (superlinear) \(f\) on a convex cone, for normed linear spaces and for various functionals defined for positive definite matrices.
scientific article; zbMATH DE number 871079