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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability: A Survey | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 386715 9 pages doi 10.1155 2008 386715 Research Article Some Equivalent Forms of the Arithematic-Geometric Mean Inequality in Probability A Survey Cheh-Chih Yeh 1 Hung-Wen Yeh 2 and Wenyaw Chan3 1 Department of Information Management Lunghwa University of Science and Technology Kueishan Taoyuan Taoyuan County 33306 Taiwan 2 Department of Biostatistics University of Kansas Kansas City KS 66160 USA 3 Division of Biostatistics University ofTexas-Health Science Center at Houston Houston TX 77030 USA Correspondence should be addressed to Cheh-Chih Yeh chehchihyeh@yahoo.com.tw Received 5 December 2007 Revised 10 April 2008 Accepted 24 June 2008 Recommended by Jewgeni Dshalalow We link some equivalent forms of the arithmetic-geometric mean inequality in probability and mathematical statistics. Copyright 2008 Cheh-Chih Yeh et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The arithmetic-geometric mean inequality in short AG inequality has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove that the AG inequality is equivalent to some other renowned inequalities by using probabilistic arguments. Among such inequalities are those of Jensen Holder Cauchy Minkowski and Lyapunov to name just a few. 2. The equivalent forms Let X be a random variable we define Er X E X r 1 r y xp E ln X if r f 0 if r 0 2.1 where EX denotes the expected value of X. 2 Journal of Inequalities and Applications Throughout this paper let n be a positive integer
