Jensen Functionals on Time Scales for Several Variables

We define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities. 1. Introduction Jensen's inequality is well known in analysis and many other areas of mathematics. Most of the classical inequalities can be obtained by using the Jensen inequality. For time scale theory, Jensen's inequality for one variable is obtained by Agarwal et al. [1], and now there are various extensions and generalizations of it given by many researchers (see [2–8]). In [3], it is shown that the Jensen inequality for one variable holds for time scale integrals including the Cauchy delta, Cauchy nabla, diamond- , Riemann, Lebesgue, multiple Riemann, and multiple Lebesgue integrals. Further, in [4], we give properties and applications of Jensen functionals on time scales for one variable. In this paper, we obtain the Jensen inequality for several variables and deduce Jensen functionals. We discuss several properties and applications of Jensen functionals. In the sequel, we give all the results for Lebesgue delta integrals. For other time scale integrals, as mentioned above, all those results can be obtained in a similar way. These results generalize the results given in [4] for one variable. Now, we give a brief introduction of time scale integrals; for a detailed introduction we refer to [1, 9–12]. A time scale is an arbitrary closed subset of , and time scale calculus provides unification and extension of classical results. For example, when , the time scale integral is an ordinary integral, and when , the time scale integral becomes a sum. In [10, Chapter 5], the Lebesgue integral is introduced: let be a time scale interval defined by where with . Let be the Lebesgue -measure on . Suppose is a -measurable function. Then the Lebesgue -integral of on is denoted by All theorems of the general Lebesgue integration theory, including the Lebesgue dominated convergence theorem, hold also for Lebesgue -integrals on . Now, we give some properties of Lebesgue -integrals and state Jensen's inequality and H？lder's inequality for Lebesgue -integrals. Throughout this paper, denotes a time scale interval otherwise is specified. Theorem 1 (see [3, Theorem 3.2]). If and are -integrable functions on , then Theorem 2 (see [3, Theorem 4.2]). Assume is convex, where is an interval. Suppose is -integrable. Moreover, let be nonnegative and -integrable such that . Then Theorem 3 (see [3,