Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

Judit Makó; Zsolt Páles

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 1017-1041
  • ISSN: 2391-5455

Abstract

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The connection between the functional inequalities f x + y 2 ⩽ f x + f y 2 + α J x - y , x , y ∈ D , and ∫ 0 1 f t x + 1 - t y ρ t d t ⩽ λ f x + 1 - λ f y + α H x - y , x , y ∈ D , is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.

How to cite

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Judit Makó, and Zsolt Páles. "Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities." Open Mathematics 10.3 (2012): 1017-1041. <http://eudml.org/doc/269223>.

@article{JuditMakó2012,
abstract = {The connection between the functional inequalities \[f\left( \{\frac\{\{x + y\}\}\{2\}\} \right) \leqslant \frac\{\{f\left( x \right) + f\left( y \right)\}\}\{2\} + \alpha \_J \left( \{x - y\} \right), x,y \in D,\] and \[\int \_0^1 \{f\left( \{tx + \left( \{1 - t\} \right)y\} \right)\rho \left( t \right)dt \leqslant \lambda f\left( x \right) + \left( \{1 - \lambda \} \right)f\left( y \right) + \alpha \_\{\rm H\} \left( \{x - y\} \right),\} x,y \in D,\] is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.},
author = {Judit Makó, Zsolt Páles},
journal = {Open Mathematics},
keywords = {Convexity; Approximate convexity; Lower and upper Hermite-Hadamard inequalities; convexity; approximate convexity; lower and upper Hermite–Hadamard inequalities},
language = {eng},
number = {3},
pages = {1017-1041},
title = {Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities},
url = {http://eudml.org/doc/269223},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Judit Makó
AU - Zsolt Páles
TI - Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 1017
EP - 1041
AB - The connection between the functional inequalities \[f\left( {\frac{{x + y}}{2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}}{2} + \alpha _J \left( {x - y} \right), x,y \in D,\] and \[\int _0^1 {f\left( {tx + \left( {1 - t} \right)y} \right)\rho \left( t \right)dt \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \alpha _{\rm H} \left( {x - y} \right),} x,y \in D,\] is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.
LA - eng
KW - Convexity; Approximate convexity; Lower and upper Hermite-Hadamard inequalities; convexity; approximate convexity; lower and upper Hermite–Hadamard inequalities
UR - http://eudml.org/doc/269223
ER -

References

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