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• W1756606122 abstract "The celebrated Turán inequalities <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis minus upper P Subscript n minus 1 Baseline left-parenthesis x right-parenthesis upper P Subscript n plus 1 Baseline left-parenthesis x right-parenthesis greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of left-bracket negative 1 comma 1 right-bracket> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>x in [-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P Subscript n Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>P_{n}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the Legendre polynomial of degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma Subscript n Superscript 2 Baseline minus gamma Subscript n minus 1 Baseline gamma Subscript n plus 1 Baseline greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma _{n}^{2} - gamma _{n-1} gamma _{n+1} geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which hold for the Maclaurin coefficients of the real entire function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=psi> <mml:semantics> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:annotation encoding=application/x-tex>psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Laguerre-Pólya class, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=psi left-parenthesis x right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts gamma Subscript n Baseline x Superscript n Baseline slash n factorial> <mml:semantics> <mml:mrow> <mml:mi>ψ<!-- ψ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>psi (x) = sum _{n=0}^{infty } gamma _{n} x^{n}/n!</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1756606122 created "2016-06-24" @default.
• W1756606122 creator A5061055854 @default.
• W1756606122 date "1998-01-01" @default.
• W1756606122 modified "2023-09-30" @default.
• W1756606122 title "Higher order Turán inequalities" @default.
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