On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function

In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the ‘two-point right focal’ problem. We use Fink’s identity and a new Abel-Gontscharoff-type Green’s function for a ‘two-point right focal’ to generalize the refinement of Jensen’s inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen’s inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed. © 2017, The Author(s).

Authors
Niaz T.1 , Khan K.A.1 , Pečarić J. 2
Publisher
Springer International Publishing
Language
English
Status
Published
Number
254
Volume
2017
Year
2017
Organizations
  • 1 Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan
  • 2 RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russian Federation
Keywords
Abel-Gontscharoff interpolating polynomial; convex function; Fink’s identity; Green function for ‘two-point right focal’ problem; Jensen’s inequality
Date of creation
19.10.2018
Date of change
19.10.2018
Short link
https://repository.rudn.ru/en/records/article/record/5984/
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