Mond-Pecaric Method in Operator Inequalities
Mond-Pecaric Method in Operator Inequalities
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发表时间：
2005
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通讯作者：
T. Furuta;J. Mićić;J. Pečarić;Y. Seo
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作者：
T. Furuta;J. Mićić;J. Pečarić;Y. Seo
第一章对正线性映射的Jensen不等式和几种类型的Kantorovich不等式中的一些基本问题作了简要回顾。给出了Mond-Peceric方法的一些基本思想和观点。第二章讨论了Jensen不等式的一般逆。用Mond-Peceric法求出边界。特别考虑了许多有趣的不平等。第三章讨论了Li-Mathias定理在归一化正线性映射中的推广，并将其应用于Mond-Peceric方法。给出了Jensen型不等式的逆的下界和上界。研究了尖锐不等式的情形。特别考虑了Jensen不等式和其他不等式的转换。在第4章中，应用前面的结果和相同的方法来得到均值的不等式。研究了正线性映射上幂算子均值的逆不等式。研究了混沌阶下幂算子均值的几个性质。给出了幂算子均值不等式的新界。第五章介绍了Kubo和Ando建立的与算子单调函数相关的算子均值理论。基于正线性映射上Jensen不等式的互补不等式，研究了与算子均值相关的Ando不等式的互补不等式。第六章应用第二章的结果和同样的方法得到了Hadamard积的不等式。然后考虑了算子和算子均值的Hadamard积上的逆不等式。观察了算子的Hadamard积的一般不等式。第七章简要介绍了古田不等式和广义古田不等式的几种应用。在第八章中，作为Mond-Peceric方法的一个应用，考虑了保留算子序和混沌序的声明。特别考虑了保持算子序和混沌序的函数的总体结果。
In Chapter 1 a very brief and rapid review of some basic topics in Jensen's inequality for positive linear maps and Kantorovich inequality for several types are given. Some basic ideas and the viewpoints of the Mond-Peceric method are given. In Chapter 2 general converses of Jensen's inequality are considered. The Mond-Peceric method is used to obtain the bounds. Many interesting inequalities are particularly considered. In Chapter 3 a generalization of a theorem of Li-Mathias for the normalized positive linear maps as an application of the Mond-Peceric method is considered. Lower and upper bounds in converses of Jensen's type inequalities are given. The cases of the sharp inequalities are investigated. The conversions of Jensen's inequality and other inequalities are particularly considered. In Chapter 4 the previous results and the same methods are applied to obtain the inequalities for the means. Reverse inequalities of power operator means on positive linear maps are studied. Several properties of power operator means under the chaotic order are considered. New bounds in inequalities for power operator means are given. In chapter 5 the theory of operator means established by Kubo and Ando assocaiated with the operator monotone functions is introduced. Based on complementary inequalities to Jensen's inequalities on positive linear maps, complementary inequalities to Ando's inequalities assocaiated with operator means are studied. In Chapter 6 the results and the same methods in the chapter 2 are applied to obtain the inequalities for the Hadamard product. Then the reverses inequalities on the Hadamard product of operators and operator means are considered. General inequalities for the Hadamard product of operators are observed. In chapter 7 a brief survey of several applications of both Furuta inequality and generalized Furuta inequality is given. In Chapter 8 the claims preserving the operator order and the chaotic order are considered as an application of the Mond-Peceric method. The overall results on the functions which preserve the operator order and the chaotic order are particularly considered.