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### Pregled bibliografske jedinice broj: 728332

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 Autori: Klaričić Bakula, Milica; Pečarić, Josip; Perić, Jurica Naslov: An improvement of the converse Jensen inequality Izvornik: Conference on Inequalities and Applications'14: Book of Abstracts / Páles, Zsolt ; Gilányi, Attila (ur.). - Debrecen : Institute of Mathematics, University of Debrecen , 2014. 38-38. Skup: Conference on Inequalities and Applications'14 Mjesto i datum: Hajdúszoboszló, Mađarska, 07.-13. 2014. Ključne riječi: Converse Jensen's inequality; Converse Jessen's inequality Sažetak: An improvement of the converse Jensen inequalityJ. Pečarić and J. Perić Let I be an interval in R and f:I→ℝ a convex function on I. If x=(x₁, …, x_{;n};) is any n-tuple in Iⁿ and p=(p₁, …, p_{;n};) a nonnegative n-tuple such that P_{;n};=∑_{;i=1};ⁿp_{;i};>0, then the well known Jensen's inequality f((1/(P_{;n};))∑_{;i=1};ⁿp_{;i};x_{;i};)≤(1/(P_{;n};))∑_{;i=1};ⁿp_{;i};f(x_{;i};) jen holds. If f is strictly convex then (jen) is strict unless x_{;i};=c for all i∈{;j:p_{;j};>0};. Jensen's inequality is probably the most important of all inequalities: it has many applications in mathematics and statistics and some other well known inequalities are its special cases (such as Cauchy's inequality, Hölder's inequality, A-G-H inequalities, etc.). Strongly related to Jensen's inequality is so called the converse Jensen inequality (1/(P_{;n};))∑_{;i=1};ⁿp_{;i};f(x_{;i};)≤((M-x)/(M-m))f(m)+((x-m)/(M-m))f(M), cjen which holds when f:I→ℝ is a convex function on I, [m, M]⊂I, -∞jen) and x=(1/P_{;n};)∑_{;i=1};ⁿp_{;i};x_{;i};. If f is strictly convex then (cjen) is strict unless x_{;i};∈{;m, M}; for all i∈{;j:p_{;j};>0};. Here we give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results. Vrsta sudjelovanja: Pozvano Vrsta prezentacije u zborniku: Sažetak Vrsta recenzije: Međunarodna recenzija Projekt / tema: Potpora 4.1.2.12. Izvorni jezik: ENG Kategorija: Znanstveni Znanstvena područja: Matematika Upisao u CROSBI: Milica Klaričić Bakula (milica@pmfst.hr), 9. Stu. 2014. u 11:14 sati   Sadržaj objavljen na bib.irb.hr (osim ako je drugačaije navedeno) podložan je pravilima Creative Commons licence Copyright © 1997-2015. IRB. Pripremili: Ivo Batistić i Jadranka Stojanovski. Dizajn: Studio8. Software: postgresql 