We apply the theory of isotonic linear functionals to derive a series of known inequalities, extensions of known inequalities, and new inequalities in the theory of dynamic equations on time scales.
Abstract and Applied AnalysisThe monograph 2 contains numerous classical inequalities that are proved for the so-called isotonic linear functionals. Since the time-scale integral is in fact an isotonic linear functional, the results from 2 can be applied to this setting. Our work shows that it is not necessary to prove such kinds of inequalities "from scratch" in the time-scale setting as they can all be obtained easily from well-known inequalities for isotonic linear functionals.The setup of this paper is as follows. In the next section, we review some known results from the literature concerning Jensen's inequality on time-scale. Section 3 contains the definition of an isotonic linear functional and the confirmations that the time-scale Cauchy delta, Cauchy nabla, α-diamond, multiple Riemann, and multiple Lebesgue integrals all are indeed isotonic linear functionals. Section 4 then is devoted to the time-scale Jensen's inequality and some of its generalizations. Some converses of Jensen's inequality in the form of time-scale Hermite-Hadamard's inequality and generalizations of it are contained in Section 5. Section 6 presents the multidimensional time-scale versions of Hölder's and Cauchy-Schwarz's inequality, followed in Section 7 by Minkowski's inequality. Section 8 is concerned with Dresher's inequality, and Section 9 offers time-scale versions of Aczél's and Popoviciu's inequalities. Section 10 contains Bellman's inequality and Section 11 deals with the Diaz-Metcalf inequality and some consequences. Five further converses of Jensen's inequality are contained in the final Section 12.
In this paper, we introduce the notion of exponentially p-convex function and exponentially s-convex function in the second sense. We establish several Hermite-Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in second sense. The present investigation is an extension of several well known results.
In this paper, we obtain the Hermite–Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite–Hadamard inequality via the Riemann–Liouville fractional integral.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.