TAILIEUCHUNG - Popoviciu type inequalities via Green function and Taylor polynomial
The well-known Taylor polynomial is used to construct the identities coming from Popovic type inequalities for convex functions via the Green function. | Turk J Math (2016) 40: 333 – 349 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Popoviciu type inequalities via Green function and Taylor polynomial 1 ˇ ´3 Saad Ihsan BUTT1 , Khuram Ali KHAN2,∗, Josip PECARI C Department of Mathematics, COMSATS, Institute of Information Technology, Lahore, Pakistan 2 Department of Mathematics, University of Sargodha, Sargodha, Pakistan 3 Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia Received: • • Accepted/Published Online: Final Version: Abstract: The well-known Taylor polynomial is used to construct the identities coming from Popoviciu type inequalities ˇ for convex functions via the Green function. The bounds for the new identities are found using the Cebyˇ sev functional to develop the Gr¨ uss and Ostrowski type inequalities. Further, more exponential convexity together with Cauchy means is presented for linear functionals associated with the obtained inequalities. ˇ Key words: Popoviciu inequality, Taylor formula, Green function, Cebyˇ sev functional, Gr¨ uss inequality, Ostrowski inequality, exponential convexity, Cauchy mean 1. Introduction and preliminary results The theory of convex functions has experienced a rapid development. This can be attributed to several causes: first, so many areas in modern analysis directly or indirectly involve the application of convex functions; second, convex functions are closely related to the theory of inequalities and many important inequalities are consequences of the applications of convex functions (see [10]). Divided differences are found to be very helpful when we are dealing with functions having different degrees of smoothness. The following definition of divided difference is given in [10, p. 14]. Definition 1 The m th-order divided difference of a function f : [a, b] → R at mutually distinct points x0 , ., xm ∈ [a, b] is defined .
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