Convexity and matrix means
Convexity and matrix means
复制标题
凸性和矩阵均值
DOI：
10.1016/j.laa.2016.06.027
复制
发表时间：
2016
影响因子：
1.1
通讯作者：
M. Sababheh
中科院分区：
文献类型：
--
作者：
M. Sababheh
在本文中，我们提出了凸函数的一些均值不等式，这些均值导致了处理数字和矩阵的算术、几何和调和均值的一些广义不等式。我们的第一个主要不等式是 (ν τ) λ≤((1− ν) f (0)+ ν f (1)) λ− f λ (ν)((1− τ) f (0)+ τ f (1)) λ− f λ (τ)≤(1− ν 1− τ) λ，对于凸函数 f，当 λ≥ 1 且 0< ν≤ τ< 1 时。此外，当 λ= 1、该不等式对于算子凸函数有效。然后通过选择适当的凸函数，我们得到一定的矩阵不等式。特别是，我们使用实数和算子凸性获得了算子的几个混合平均不等式。我们的讨论将带来矩阵 Heinz 和 Hölder 不等式的新乘法改进和逆运算、新的和改进的迹以及行列式不等式。这项工作的意义在于它的一般处理，其中凸性是唯一需要的属性。
In this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic, geometric and harmonic means for numbers and matrices. Our first main inequality will be (ν τ) λ≤((1− ν) f (0)+ ν f (1)) λ− f λ (ν)((1− τ) f (0)+ τ f (1)) λ− f λ (τ)≤(1− ν 1− τ) λ, for the convex function f, when λ≥ 1 and 0< ν≤ τ< 1. Moreover, when λ= 1, the inequality will be valid for operator convex functions. Then by selecting an appropriate convex function, we obtain certain matrix inequalities. In particular, we obtain several mixed mean inequalities for operators using real and operator convexity. Our discussion will lead to new multiplicative refinements and reverses of the Heinz and Hölder inequalities for matrices, new and refined trace and determinant inequalities. The significance of this work is its general treatment, where convexity is the only needed property.