Approximate convexity with respect to a subfield View Full Text


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Article Info

DATE

2017-03-20

AUTHORS

Z. Boros, N. Nagy

ABSTRACT

Let F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}}$$\end{document} be a subfield of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} and X be a linear space over F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}}$$\end{document}. Let D⊆X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ D\subseteq X }$$\end{document} be a nonempty F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}}$$\end{document}-convex set, D∗:=D-D:={x-y:x,y∈D}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ D^*:=D-D:=\{x-y : x,y\in D\} }$$\end{document}, and α:D∗→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha \colon {D^* \rightarrow \mathbb{R}}}$$\end{document} be a nonnegative even function. The function f:D→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ f \colon {D\rightarrow \mathbb{R}}}$$\end{document} is called (α,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\alpha,\mathbb{F})}$$\end{document}-convex, if it satisfies the inequality f(tx+(1-t)y)≤tf(x)+(1-t)f(y)+tα((1-t)(x-y))+(1-t)α(t(y-x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) + t\alpha((1-t)(x-y))+(1-t)\alpha(t(y-x))}\end{aligned}$$\end{document}for all x,y∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x,y\in D}$$\end{document} and for all t∈F∩[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t\in \mathbb{F}\cap [0,1]}$$\end{document}. In this paper we characterize (α,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\alpha,\mathbb{F})}$$\end{document}-convex functions by comparison of modified difference ratios and support properties. If α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha}$$\end{document} satisfies some additional conditions, we obtain the differentiability of (α,F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\alpha,\mathbb{F})}$$\end{document}-convex functions in the appropriate sense. More... »

PAGES

464-472

References to SciGraph publications

  • 1915-12. Zur Theorie der konvexen Funktionen in MATHEMATISCHE ANNALEN
  • 2010-09-16. Remarks on strongly convex functions in AEQUATIONES MATHEMATICAE
  • 1906-12. Sur les fonctions convexes et les inégalités entre les valeurs moyennes in ACTA MATHEMATICA
  • 2010-08-12. Optimality estimations for approximately midconvex functions in AEQUATIONES MATHEMATICAE
  • Identifiers

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    http://scigraph.springernature.com/pub.10.1007/s10474-017-0701-y

    DOI

    http://dx.doi.org/10.1007/s10474-017-0701-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1084716127


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    25 difference ratio
    26 differentiability
    27 function
    28 function f
    29 inequality
    30 linear space
    31 nonnegative
    32 paper
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