§18.14 Inequalities

Permalink:: http://dlmf.nist.gov/18.14

§18.14(i) Upper Bounds
§18.14(ii) Turan-Type Inequalities
§18.14(iii) Local Maxima and Minima

§18.14(i) Upper Bounds

Notes:: For (18.14.1) and (18.14.2) see Szegö (1975, Theorem 7.32.1). For (18.14.3) see Chow et al. (1994). For (18.14.4), (18.14.5), and (18.14.6) see Szegö (1975, Theorem 7.33.1). For (18.14.7) see Lorch (1984). For (18.14.8) see Koornwinder (1977, Remark 4.1). For (18.14.9) see Szász (1951).
Keywords:: classical orthogonal polynomials, inequalities
Permalink:: http://dlmf.nist.gov/18.14.i

¶ Jacobi

Keywords:: Jacobi polynomials, inequalities

18.14.1 $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)|\leq\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1\right)=\frac{\left(\alpha+1\right)_{{n}}}{n!},$ $-1\leq x\leq 1$ , $\alpha\geq\beta>-1$ , $\alpha\geq-\tfrac{1}{2},$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.1
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E1
Encodings:: TeX, pMML, png

18.14.2 $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)|\leq|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(-1\right)|=\frac{\left(\beta+1\right)_{{n}}}{n!},$ $-1\leq x\leq 1$ , $\beta\geq\alpha>-1$ , $\beta\geq-\tfrac{1}{2}$ .

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.1
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E2
Encodings:: TeX, pMML, png

18.14.3 $\left(\tfrac{1}{2}(1-x)\right)^{{\frac{1}{2}\alpha+\frac{1}{4}}}\left(\tfrac{1}{2}(1+x)\right)^{{\frac{1}{2}\beta+\frac{1}{4}}}|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)|\leq\frac{\mathop{\Gamma\/}\nolimits\!\left(\max(\alpha,\beta)+n+1\right)}{\pi^{{\frac{1}{2}}}n!\left(n+\tfrac{1}{2}(\alpha+\beta+1)\right)^{{\max(\alpha,\beta)+\frac{1}{2}}}},$ $-1\leq x\leq 1$ , $-\tfrac{1}{2}\leq\alpha\leq\tfrac{1}{2}$ , $-\tfrac{1}{2}\leq\beta\leq\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $!$ : $n!$ : factorial, $(a,b)$ : open interval, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E3
Encodings:: TeX, pMML, png

¶ Ultraspherical

Keywords:: inequalities, ultraspherical polynomials

18.14.4 $|\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)|\leq\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(1\right)=\frac{\left(2\lambda\right)_{{n}}}{n!},$ $-1\leq x\leq 1$ , $\lambda>0$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.2
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E4
Encodings:: TeX, pMML, png

18.14.5 $|\mathop{C^{{(\lambda)}}_{{2m}}\/}\nolimits\!\left(x\right)|\leq|\mathop{C^{{(\lambda)}}_{{2m}}\/}\nolimits\!\left(0\right)|=\left|\frac{\left(\lambda\right)_{{m}}}{m!}\right|,$ $-1\leq x\leq 1$ , $-\tfrac{1}{2}<\lambda<0$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $m$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.2
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E5
Encodings:: TeX, pMML, png

18.14.6 $|\mathop{C^{{(\lambda)}}_{{2m+1}}\/}\nolimits\!\left(x\right)|<\frac{-2\left(\lambda\right)_{{m+1}}}{\left((2m+1)(2\lambda+2m+1)\right)^{{\frac{1}{2}}}m!},$ $-1\leq x\leq 1$ , $-\tfrac{1}{2}<\lambda<0$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $m$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E6
Encodings:: TeX, pMML, png

18.14.7 ${(n+\lambda)^{{1-\lambda}}(1-x^{2})^{{\frac{1}{2}\lambda}}|\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)|<\frac{2^{{1-\lambda}}}{\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)}},$ $-1\leq x\leq 1$ , $0<\lambda<1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E7
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials, inequalities

18.14.8 $e^{{-\frac{1}{2}x}}\left|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)\right|\leq\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(0\right)=\frac{\left(\alpha+1\right)_{{n}}}{n!},$ $0\leq x<\infty$ , $\alpha\geq 0$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.13
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E8
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Chebyshev polynomials, Hermite polynomials, inequalities

18.14.9 $\frac{1}{(2^{n}n!)^{{\frac{1}{2}}}}e^{{-\frac{1}{2}x^{2}}}|\mathop{H_{{n}}\/}\nolimits\!\left(x\right)|\leq 1,$ $-\infty<x<\infty$ .

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.14.17 ((version here is sharpened))
Referenced by:: §18.14(i)
Permalink:: http://dlmf.nist.gov/18.14.E9
Encodings:: TeX, pMML, png

For further inequalities see Abramowitz and Stegun (1964, §22.14).

§18.14(ii) Turan-Type Inequalities

Notes:: For (18.14.10) see Szegö (1948). For (18.14.11) see Gasper (1972). For (18.14.12), (18.14.13) see Skovgaard (1954).
Keywords:: classical orthogonal polynomials, inequalities
Permalink:: http://dlmf.nist.gov/18.14.ii

¶ Legendre

Keywords:: Legendre polynomials, inequalities

18.14.10 $(\mathop{P_{{n}}\/}\nolimits\!\left(x\right))^{2}\geq\mathop{P_{{n-1}}\/}\nolimits\!\left(x\right)\mathop{P_{{n+1}}\/}\nolimits\!\left(x\right),$ $-1\leq x\leq 1$ .

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(ii)
Permalink:: http://dlmf.nist.gov/18.14.E10
Encodings:: TeX, pMML, png

¶ Jacobi

Keywords:: Jacobi polynomials, inequalities

Let $R_{n}(x)=\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)/\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1\right)$ . Then

18.14.11 $(R_{n}(x))^{2}\geq R_{{n-1}}(x)R_{{n+1}}(x),$ $-1\leq x\leq 1$ , $\beta\geq\alpha>-1$ .

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(ii)
Permalink:: http://dlmf.nist.gov/18.14.E11
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials, inequalities

18.14.12 $(\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right))^{2}\geq\mathop{L^{{(\alpha)}}_{{n-1}}\/}\nolimits\!\left(x\right)\mathop{L^{{(\alpha)}}_{{n+1}}\/}\nolimits\!\left(x\right),$ $0\leq x<\infty$ , $\alpha\geq 0$ .

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(ii)
Permalink:: http://dlmf.nist.gov/18.14.E12
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials, inequalities

18.14.13 $(\mathop{H_{{n}}\/}\nolimits\!\left(x\right))^{2}\geq\mathop{H_{{n-1}}\/}\nolimits\!\left(x\right)\mathop{H_{{n+1}}\/}\nolimits\!\left(x\right),$ $-\infty<x<\infty$ .

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(ii)
Permalink:: http://dlmf.nist.gov/18.14.E13
Encodings:: TeX, pMML, png

§18.14(iii) Local Maxima and Minima

Notes:: For (18.14.14)–(18.14.19) see Szegö (1975, discussion following Theorem 7.32.1). For (18.14.20) see Szász (1950). For (18.14.21)–(18.14.24) see Szegö (1975, Theorem 7.6.1). For the last statement about the successive maxima of $|\mathop{H_{{n}}\/}\nolimits\!\left(x\right)|$ see Szegö (1975, Theorem 7.6.3).
Keywords:: classical orthogonal polynomials, inequalities, local maxima and minima
Permalink:: http://dlmf.nist.gov/18.14.iii

¶ Jacobi

Keywords:: Chebyshev polynomials, Jacobi polynomials, inequalities, local maxima and minima

Let the maxima $x_{{n,m}}$ , $m=0,1,\dots,n$ , of $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)|$ in $[-1,1]$ be arranged so that

18.14.14 $-1=x_{{n,0}}<x_{{n,1}}<\cdots<x_{{n,n-1}}<x_{{n,n}}=1.$

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(iii)
Permalink:: http://dlmf.nist.gov/18.14.E14
Encodings:: TeX, pMML, png

When $(\alpha+\tfrac{1}{2})(\beta+\tfrac{1}{2})>0$ choose $m$ so that

18.14.15 $x_{{n,m}}\leq(\beta-\alpha)/(\alpha+\beta+1)\leq x_{{n,m+1}}.$

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E15
Encodings:: TeX, pMML, png

Then

18.14.16

$|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|>\cdots>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,m}}\right)|,$

$|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n}}\right)|>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-1}}\right)|>\cdots>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,m+1}}\right)|,$ $\alpha>-\tfrac{1}{2},\beta>-\tfrac{1}{2}.$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: Figure 18.4.2, Figure 18.4.2
Permalink:: http://dlmf.nist.gov/18.14.E16
Encodings:: TeX, TeX, pMML, pMML, png, png

18.14.17

$|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|<\cdots<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,m}}\right)|,$

$|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n}}\right)|<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-1}}\right)|<\cdots<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,m+1}}\right)|,$ $-1<\alpha<-\tfrac{1}{2},-1<\beta<-\tfrac{1}{2}.$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E17
Encodings:: TeX, TeX, pMML, pMML, png, png

Also,

18.14.18 $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|<\cdots<|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n}}\right)|,$ $\alpha\geq-\tfrac{1}{2}$ , $-1<\beta\leq-\tfrac{1}{2}$ ,

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E18
Encodings:: TeX, pMML, png

18.14.19 $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|>\cdots>|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n}}\right)|,$ $\beta\geq-\frac{1}{2}$ , $-1<\alpha\leq-\tfrac{1}{2}$ ,

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(iii)
Permalink:: http://dlmf.nist.gov/18.14.E19
Encodings:: TeX, pMML, png

except that when $\alpha=\beta=-\tfrac{1}{2}$ (Chebyshev case) $|\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,m}}\right)|$ is constant.

¶ Szegö–Szász Inequality

Keywords:: Jacobi polynomials, Szegö–Szász inequality, inequalities, local maxima and minima

18.14.20 $\left|\frac{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-m}}\right)}{\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1\right)}\right|>\left|\frac{\mathop{P^{{(\alpha,\beta)}}_{{n+1}}\/}\nolimits\!\left(x_{{n+1,n-m+1}}\right)}{\mathop{P^{{(\alpha,\beta)}}_{{n+1}}\/}\nolimits\!\left(1\right)}\right|,$ $\alpha=\beta>-\tfrac{1}{2}$ , $m=1,2,\dots,n$ .

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: ¶ ‣ §18.14(iii), §18.14(iii), §18.39(ii)
Permalink:: http://dlmf.nist.gov/18.14.E20
Encodings:: TeX, pMML, png

For extensions of (18.14.20) see Askey (1990) and Wong and Zhang (1994a, b).

¶ Laguerre

Keywords:: Laguerre polynomials, inequalities, local maxima and minima

Let the maxima $x_{{n,m}}$ , $m=0,1,\dots,n-1$ , of $|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)|$ in $[0,\infty)$ be arranged so that

18.14.21 $0=x_{{n,0}}<x_{{n,1}}<\dots<x_{{n,n-1}}<x_{{n,n}}=\infty.$

Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(iii)
Permalink:: http://dlmf.nist.gov/18.14.E21
Encodings:: TeX, pMML, png

When $\alpha>-\tfrac{1}{2}$ choose $m$ so that

18.14.22 $x_{{n,m}}\leq\alpha+\tfrac{1}{2}\leq x_{{n,m+1}}.$

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E22
Encodings:: TeX, pMML, png

Then

18.14.23

$|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|>|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|>\cdots>|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,m}}\right)|,$

$|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-1}}\right)|>|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-2}}\right)|>\cdots>|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,m+1}}\right)|.$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.14.E23
Encodings:: TeX, TeX, pMML, pMML, png, png

Also, when $\alpha\leq-\tfrac{1}{2}$

18.14.24 $|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,0}}\right)|<|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,1}}\right)|<\dots<|\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x_{{n,n-1}}\right)|.$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.14(iii)
Permalink:: http://dlmf.nist.gov/18.14.E24
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials, classical orthogonal polynomials, inequalities, local maxima and minima

The successive maxima of $|\mathop{H_{{n}}\/}\nolimits\!\left(x\right)|$ form a decreasing sequence for $x\leq 0$ , and an increasing sequence for $x\geq 0$ .

§18.14 Inequalities

Contents

§18.14(i) Upper Bounds

¶ Jacobi

¶ Ultraspherical

¶ Laguerre

¶ Hermite

§18.14(ii) Turan-Type Inequalities

¶ Legendre

¶ Jacobi

¶ Laguerre

¶ Hermite

§18.14(iii) Local Maxima and Minima

¶ Jacobi

¶ Szegö–Szász Inequality

¶ Laguerre

¶ Hermite