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Báo cáo hóa học: " Exponential convexity of Petrović and related functional"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Exponential convexity of Petrović and related functional | Butt et al. Journal of Inequalities and Applications 2011 2011 89 http www.journalofinequalitiesandapplications.eom content 2011 1 89 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Exponential convexity of Petrovic and related functional Saad I Butt 1 Josip Pecaric1 2 and Atiq Ur Rehman3 Correspondence saadihsanbutt@gmail.com 1Abdus Salam School of Mathematical Sciences GC University Lahore Pakistan Full list of author information is available at the end of the article Springer Abstract We consider functionals due to the difference in Petrovic and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions. We construct positive semi-definite matrices generated by these functionals and give some related results. At the end we give some examples. Keywords convex functions divided difference exponentially convex functionals log-convex functions positive semi-definite 1 Introduction First time exponentially convex functions are introduced by Bernstein 1 . Independently of Bernstein but some what later Widder 2 introduced these functions as a sub-class of convex functions in a given interval a b and denoted this class by Wa b. After the initial development there is a big gap in time before applications and examples of interest were constructed. One of the reasons is that aside from absolutely monotone functions and completely monotone functions as special classes of exponentially convex functions there is no operative criteria to recognize exponential convexity of functions. Definition 1. 3 p. 373 A function f a b R is exponentially convex if it is continuous and E Mjf Xi Xj 0 1 i j 1 for all n e N and all choices fi e R and xi Xj e a b 1 i j n. Proposition 1.1. Let f a b R. The following propositions are equivalent. i f is exponentially convex. ii f is continuous and - Í Xi xj Y j 0 i j 1 for every i e R and every Xi e a b 1 i n. 2011 Butt et al licensee