In this paper, G-majorization inequalities are proven for the gradients of two Gateaux differentiable G-increasing functions defined on a real linear space endowed with the structure of an Eaton triple. Such inequalities can be interpreted as a monotonicity property of certain operators and functionals with respect to a special preorder on the class of all G-increasing functions. The notion of a c-strongly G-increasing function is introduced and used to illustrate the derived inequalities. The obtained results are also applied for matrix Eaton triples connected with the eigenvalues map on the space of Hermitian matrices as well as with the singular values map on the space of complex matrices.

在本文中，证明了在赋予伊顿三元组结构的实线性空间上定义的两个 Gateaux 可微G增函数的梯度的G主要化不等式。这种不等式可以解释为某些运算符和泛函的单调性关于所有G增加函数类的特殊预序。引入了c -强G -增加函数的概念，并用于说明导出的不等式。所得结果也适用于与厄米特矩阵空间上的特征值映射以及复数矩阵空间上的奇异值映射相连的矩阵伊顿三元组。