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http://functions.wolfram.com/06.05.29.0007.01
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Inequality[n^(1 - x), Less, Gamma[n + 1]/Gamma[n + x], Greater,
Exp[(1 - x) PolyGamma[n + 1]]] /; Element[n, Integers] && n > 0 &&
Element[x, Reals] && 0 < x < 1
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Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List["1", "-", "x"]]], "<", FractionBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["n", "+", "x"]], "]"]]], ">", RowBox[List["Exp", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["0", "<", "x", "<", "1"]]]]]]]]
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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> <msup> <mi> n </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> </msup> <mo> < </mo> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> > </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Inequality </ci> <apply> <power /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <lt /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <gt /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List["1", "-", "x"]]], "<", FractionBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["n", "+", "x"]], "]"]]], ">", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "x"]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["0", "<", "x", "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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