论文标题
凸功能的新处理
A new treatment of convex functions
论文作者
论文摘要
凸功能在数学不平等领域中起了重要作用。在本文中，我们介绍了与凸性有关的新概念，该概念证明了何时该功能比另一个函数更凸了。 特别是，我们将所谓的$ g- $ convexity定义为$ \ log- $ covexity的概括。然后，我们证明$ g- $凸功能在某些已知的不平等现象中具有更好的估计，例如Hermite-Hadard不平等，凸功能的超级添加性，多数不平等和某些手段不平等。 与此密切相关的是，我们将凸的索引定义为``函数是多少函数''的量度。 最终将介绍包括希尔伯特太空运营商，矩阵和熵的应用程序。
Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $\log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of ``how much the function is convex". Applications including Hilbert space operators, matrices and entropies will be presented in the end.
