Mathematical Sciences: Some Elementary Inequalities

数学分析的基础是实数系统，而实数系统又依赖于完备性和极限的概念。在理论和实践中，实数的估计和逼近都是一个基本的概念，对所有形式的科学和工程工作都是至关重要的。有几个著名的不平等--例如，那些带有柯西、施瓦茨、明可夫斯基和霍尔德的名字--很久以前就被发现了，它们是一般不平等理论的基础。当一个新的不等式被发现时，这是一个罕见而重要的事件，因为这样的发生肯定预示着重要的和以前没有被怀疑或无法获得的应用。班尼特教授最近发现了一些新的基本不平等，这一点令人羡慕。它们解决了优势论中的一些突出问题。此外，这项工作使Bennett教授能够猜测他的不等式的更强大的扩展，这反过来将在非常基本的学科如函数论、算子论、特征值分布、正交级数和矩阵理论中开辟新的研究方向。班尼特教授将继续这一独特而重要的研究计划。

The foundation upon which mathematical analysis rests is the real number system, which in turn depends upon the concept of completeness and limits. Both in theory and practice, estimation and approximation of real numbers is a fundamental concept, crucial to all forms of scientific and engineering endeavor. There are a few famous inequalities -- those that bear the names Cauchy, Schwarz, Minkowski, and Holder, for example -- that were discovered long ago, and which are basic to the general theory of inequalities. It is a rare and important event when a new inequality is discovered, for such an occurence surely foreshadows important and previously unsuspected or unobtainable applications. Professor Bennett has the enviable distinction of having recently discovered some new fundamental inequalities. They have led to solution of a number of outstanding problems in the theory of majorization. Moreover, this work has allowed Professor Bennett to conjecture still more powerful extensions of his inequalities, which in turn would open new research directions in very basic subjects such as function theory, operator theory, eigenvalue distribution, orthogonal series, and matrix theory. Professor Bennett will continue this unique and important research program.