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: A new interpretation of Jensen’s inequality and geometric properties of -means | Nakasuji et al. Journal of Inequalities and Applications 2011 2011 48 http content 2011 1 48 RESEARCH 3 Journal of Inequalities and Applications a SpringerOpen Journal Open Access A new interpretation of Jensen s inequality and geometric properties of -means Yasuo Nakasuji1 Keisaku Kumahara1 and Sin-Ei Takahasi2 Correspondence sin-ei@emperor. 2Toho University Yamagata University Professor Emeritus Chiba 273-0866 Japan Full list of author information is available at the end of the article Abstract We introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function Ộ. Such a mean is called a -mean of f and written by Mf We first give a new interpretation of Jensen s inequality by -mean. Next as an application we consider some geometric properties of Mf for example refinement strictly monotone increasing continuous -mean path convexity etc. Mathematics Subject Classification 2000 Primary 26E60 Secondary 26B25 26B05. Keywords Jensen s inequality Mean Refinement Convexity Concavity 1. Introduction We are interested in means of real-valued measurable functions induced by strictly monotone functions. These means are somewhat different from continuously differentiable means . C1-means introducing by Fujii et al. 1 but they include many known numerical means. Here we first give a new interpretation of Jensen s inequality by such a mean and we next consider some geometric properties of such means as an application of it. Throughout the paper we denote by ít ự I and f a probability space an interval of R and a real-valued measurable function on í with f m e I for almost all m e í respectively. Let C I be the real linear space of all continuous real-valued functions defined on I. Let C m I resp. C-JI be the set of all Ộ e C I which is strictly monotone increasing resp. decreasing on I. Then C m I resp. C-m I is a positive resp. negative cone of C I . Put Csm
