Gamma[Product[Subscript[x, k]^Subscript[p, k], {k, 1, n}]] >= Product[Gamma[Subscript[x, k]]^Subscript[p, k], {k, 1, n}] /; Subscript[p, k] > 0 && Inequality[0, Less, Subscript[x, k], LessEqual, a] && 1 <= k <= n && PolyGamma[a] + a PolyGamma[1, a] == 0 && a == ( 0.2160987453133341405108158167798198775870564925506445459749606352433903`\ 69.33465224534193 55016505581272221260167439143622018214555137493697) \[Ellipsis]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> x </mi> <mi> k </mi> <msub> <mi> p </mi> <mi> k </mi> </msub> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#8805; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> x </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <msub> <mi> x </mi> <mi> k </mi> </msub> <mo> &#8804; </mo> <mi> a </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#10869; </mo> <mrow> <mrow> <mn> 0.2160987453133341405108158167798198775870564925506445459749606352433903 </mn> <mtext> \ </mtext> <mn> 55016505581272221260167439143622018214555137493697 </mn> </mrow> <mo> &#8290; </mo> <mo> &#8230; </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <geq /> <apply> <ci> Gamma </ci> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 0 </cn> <lt /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> <leq /> <ci> a </ci> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> n </ci> </apply> <apply> <eq /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <ci> a </ci> <apply> <times /> <apply> <times /> <cn type='real'> 0.2160987453133341405108158167798198775870564925506445459749606352433903 </cn> <cn type='integer'> 55016505581272221260167439143622018214555137493697 </cn> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29