Schur \(m\)-power convexity of a class of multiplicatively convex functions and applications
Summary: We investigate the conditions under which the symmetric functions \(F_{n, k}(\mathbf{x}, r) = \prod_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} f \left((\sum_{j = 1}^k x_{i_j}^r)^{1 / r}\right)\), \(k = 1,2, \dots, n\), are Schur \(m\)-power convex for \(x \in \mathbb{R}_{+ +}^n\) and \(r > 0\). As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving the \(p\)th power mean and the arithmetic, the geometric, or the harmonic means are presented.
scientific article; zbMATH DE number 7022017
scientific article