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Báo cáo hóa học: " Extension of Hu Ke’s inequality and its applications Jing-Feng Tian"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Extension of Hu Ke’s inequality and its applications Jing-Feng Tian | 3 Journal of Inequalities and Applications a SpringerOpen Journal Tian Journal of Inequalities and Applications 2011 2011 77 http www.journalofinequalitiesandapplications.eom content 2011 1 77 RESEARCH Open Access Extension of Hu Ke s inequality and its applications Jing-Feng Tian Correspondence tianjfhxm_ncepu@yahoo.cn College of Science and Technology North China Electric Power University Baoding Hebei Province 071051 People s Republic of China Springer Abstract In this paper we extend Hu Ke s inequality which is a sharpness of Holder s inequality. Moreover the obtained results are used to improve Hao Z-C inequality and Beckenbach-type inequality that is due to Wang. Mathematics Subject Classification 2000 Primary 26D15 Secondary 26D10 Keywords integral inequality Holder s inequality Hu Ke s inequality Hao Z-C inequality Beckenbach-type inequality arithmetic-geometric mean inequality 1. Introduction The classical Holder s inequality states that if ak 0 bk 0 k 1 2 . n p 0 q 0 and 1 then p q n n n y ap p iỉ q 1 akbk - I ak bk k 1 k 1 v k 1 The inequality 1 is reversed for p 1 p 0 . For p 0 we assume that ak bk 0. The following generalization of 1 is given in 1 Theorem A. Generalized Holder inequality . Let Anj 0 2nA Ij 0 j 1 2 . k . II z 1 then Eli n X A 1 2 11 - 2 n 1 1 n As is well known Holder s inequality plays a very important role in different branches of modern mathematics such as linear algebra classical real and complex analysis probability and statistics qualitative theory of differential equations and their applications. A large number of papers dealing with refinements generalizations and applications of inequalities 1 and 2 and their series analogues in different ares of mathematics have appeared see e.g. 2-30 and the references therein . Among various refinements of 1 Hu in 13 established the following interesting theorems. 2011 Tian licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution .