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On Vector Hermite-Hadamard Differences Controlled by Their Scalar Counterparts

Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)

Abstract

We present a new, in a sense direct, proof that the system of two functional inequalities
$$\biggl \Vert F \biggl(\frac{x+y}{2} \biggr) - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{1}{y-x} \int_{x}^{y} f(t)\,dt - f \biggl(\frac{x+y}{2} \biggr) $$
and
$$\biggl \Vert \frac{F(x) + F(y)}{2} - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{f(x) + f(y)}{2} -\frac{1}{y-x} \int_{x}^{y} f(t)\,dt $$
is satisfied for functions F and f mapping an open interval I of the real line ℝ into a Banach space and into ℝ, respectively, if and only if F yields a delta-convex mapping with a control function f.
A similar result is obtained for delta-convexity of higher orders with detailed proofs given in the case of delta-convexity of the second order, i.e. when the functional inequality holds true provided that x,yI, xy.

Keywords

Hermite-Hadamard type inequalities Delta-convex map Control function Delta-convex map of higher order 

Mathematics Subject Classification

26B25 39B72 39B62 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute of MathematicsSilesian UniversityKatowicePoland
  2. 2.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia

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