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<p>We analyze properties of the <i>index inequality</i> <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6726_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="198" /> </InlineMediaObject> <EquationSource Format="TEX">\(f^{-1}(\textbf{E}f(X))\le g^{-1}(\textbf{E}g(X))\)</EquationSource> </InlineEquation>, and of related <i>shift inequality,</i> <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6726_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="237" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{E}g(a+X)\le g(a+g^{-1}(\textbf{E}g(X)))\)</EquationSource> </InlineEquation>, and give necessary and sufficient conditions for each of these inequalities to be true for any random variable&#xa0;<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2025_6726_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\ge 0\)</EquationSource> </InlineEquation>.</p>

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The index and shift inequalities and their properties

  • 期刊论文

  • 发表日期：

    2025年7月30日

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  • Alexander Gordon,
  • Stanislav Molchanov,
  • Isaac M. Sonin

摘要

We analyze properties of the index inequality \(f^{-1}(\textbf{E}f(X))\le g^{-1}(\textbf{E}g(X))\) , and of related shift inequality, \(\textbf{E}g(a+X)\le g(a+g^{-1}(\textbf{E}g(X)))\) , and give necessary and sufficient conditions for each of these inequalities to be true for any random variable  \(X\ge 0\) .