Binomial coefficients: Inequalities (formula 06.03.29.0003)

Binomial

$Mathematica Notation$

Gamma, Beta, Erf

Binomial[n,k]

Inequalities

http://functions.wolfram.com/06.03.29.0003.01

Input Form

Subscript[d, n][\[Lambda] + \[Alpha] + \[Beta]] Sum[(Subscript[d, n - k][\[Lambda]]/Subscript[d, k][\[Alpha] + \[Beta]]) Abs[Sum[Subscript[a, j] Subscript[b, k - j], {j, 0, k}]]^2, {k, 0, n}] <= Sum[(Subscript[d, n - k][\[Lambda] + \[Beta]]/ Subscript[d, k][\[Alpha]]) Abs[Subscript[a, k]]^2, {k, 0, n}] Sum[(Subscript[d, n - k][\[Lambda] + \[Alpha]]/Subscript[d, k][\[Beta]]) Abs[Subscript[b, k]]^2, {k, 0, n}] /; Subscript[d, k][\[Alpha]] == Binomial[k + \[Alpha] - 1, k] && Element[\[Alpha], Reals] && \[Alpha] > 0 && Element[\[Beta], Reals] && \[Beta] > 0 && Element[\[Lambda], Reals] && \[Lambda] >= 0 && Max[Abs[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n]}]] > 0 && Max[Abs[{Subscript[b, 1], Subscript[b, 2], \[Ellipsis], Subscript[b, n]}]] > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]", "+", "\[Beta]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", "\[Lambda]", "]"]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Alpha]", "+", "\[Beta]"]], "]"]]], SuperscriptBox[RowBox[List["Abs", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], "]"]], "2"]]]]]]], "\[LessEqual]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Beta]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["a", "k"], "]"]], "2"]]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]"]], "]"]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Beta]", "]"]]], SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["b", "k"], "]"]], "2"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], ",", "k"]], "]"]]]], "\[And]", RowBox[List["\[Alpha]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]], "\[And]", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Beta]", ">", "0"]], "\[And]", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "}"]], "]"]], "]"]], ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> d </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> β </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> β </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <mo> ≤ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> β </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> a </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> b </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> k </mi> <mo> + </mo> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> β </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> max </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> <mo> } </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> n </ci> </apply> <apply> <plus /> <ci> α </ci> <ci> β </ci> <ci> λ </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <ci> λ </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> α </ci> <ci> β </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> β </ci> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> α </ci> <ci> λ </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> β </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> α </ci> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> <apply> <in /> <ci> α </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> β </ci> <apply> <ci> SuperPlus </ci> <reals /> </apply> </apply> <apply> <in /> <ci> λ </ci> <ci> ℝ </ci> </apply> <apply> <geq /> <ci> λ </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <max /> <apply> <abs /> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> </list> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "n"], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]", "+", "\[Beta]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", "\[Lambda]", "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], " ", SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", RowBox[List["\[Alpha]", "+", "\[Beta]"]], "]"]]]]]]], "\[LessEqual]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Beta]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["a", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "k"]]], "[", RowBox[List["\[Lambda]", "+", "\[Alpha]"]], "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", SubscriptBox["b", "k"], "]"]], "2"]]], RowBox[List[SubscriptBox["d", "k"], "[", "\[Beta]", "]"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "\[Alpha]", "]"]], "\[Equal]", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", "\[Alpha]", "-", "1"]], ",", "k"]], "]"]]]], "&&", RowBox[List["\[Alpha]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Alpha]", ">", "0"]], "&&", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Beta]", ">", "0"]], "&&", RowBox[List["\[Lambda]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Lambda]", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], "]"]], "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Max", "[", RowBox[List["Abs", "[", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "}"]], "]"]], "]"]], ">", "0"]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2002-12-18