A new interpretation of Jensen's inequality and geometric properties of <b> <it>&#966;</it> </b>-means

Abstract We introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function &#966;. Such a mean is called a &#966;-mean of f and written by M&#966; (f). We first give a new interpretation of Jensen's inequality by &#966;-mean. Next, as an application, we consider some geometric properties of M&#966; (f), for example, refinement, strictly monotone increasing (continuous) &#966;-mean path, convexity, etc. Mathematics Subject Classification (2000): Primary 26E60; Secondary 26B25, 26B05.</p