Abstract
The article first shows one alternative definition of convexity in the discrete case. The correlation between barycenters, Jensen’s inequality and convexity is studied in the integral case. The Hermite-Hadamard inequality is also obtained as a consequence of a concept of barycenters. Some derived results are applied to the quasi-arithmetic means and especially to the power means.
Highlights
1 Introduction Sets with the common barycenter are observed in geometry, mechanics dealing with mass densities and probability theory in the study of random variables
We wanted to insert the quasi-arithmetic means into this implication
The main results of the paper are presented in Sections and
Summary
Sets with the common barycenter are observed in geometry, mechanics dealing with mass densities and probability theory in the study of random variables. Development and application of the theory of convex functions includes barycenters. The following result, expressed by the measure and integral, is the most commonly used. F (t) dμ(t) holds for every μ-integrable convex function f : B → R.’. The intention of this paper is still more to connect the quoted implication (in the extended form) with convex functions, in the discrete and integral case. The quoted result was observed in Banach spaces. It was assumed that A and B are bounded closed convex subsets of a Banach space E such that A ⊂ B and f : B → R is a convex function. Throughout the whole paper, we suppose that I ⊆ R is a non-degenerate interval.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
