# §18.14 Inequalities

## §18.14(i) Upper Bounds

### Jacobi

 18.14.1 $|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq P^{(\alpha,\beta)}_{n}\left(1\right% )=\frac{{\left(\alpha+1\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\alpha\geq\beta>-1$, $\alpha\geq-\tfrac{1}{2}$,
 18.14.2 $|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq|P^{(\alpha,\beta)}_{n}\left(-1% \right)|=\frac{{\left(\beta+1\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\beta\geq\alpha>-1$, $\beta\geq-\tfrac{1}{2}$.
 18.14.3 $\left(\tfrac{1}{2}(1-x)\right)^{\frac{1}{2}\alpha+\frac{1}{4}}\left(\tfrac{1}{% 2}(1+x)\right)^{\frac{1}{2}\beta+\frac{1}{4}}|P^{(\alpha,\beta)}_{n}\left(x% \right)|\leq\frac{\Gamma\left(\max(\alpha,\beta)+n+1\right)}{{\pi}^{\frac{1}{2% }}n!\left(n+\tfrac{1}{2}(\alpha+\beta+1)\right)^{\max(\alpha,\beta)+\frac{1}{2% }}},$ $-1\leq x\leq 1$, $-\tfrac{1}{2}\leq\alpha\leq\tfrac{1}{2}$, $-\tfrac{1}{2}\leq\beta\leq\tfrac{1}{2}$.
 18.14.3_5 $\left(\tfrac{1}{2}(1+x)\right)^{\beta/2}\left|P^{(\alpha,\beta)}_{n}\left(x% \right)\right|\leq P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left(\alpha+1% \right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\alpha,\beta\geq 0$. ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Proved: Carnicer et al. (2020, Theorem 2)(proved) Proof sketch: The case $\alpha=0$ follows from the result on monotonic weight functions in §18.2(xii). Referenced by: §18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) Section 18.14 Permalink: http://dlmf.nist.gov/18.14.E3_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

Equations (18.14.3) and (18.14.3_5) are Bernstein-type inequalities. For further inequalities of this type see Koornwinder et al. (2018, §1) and references given there.

### Ultraspherical

 18.14.4 $|C^{(\lambda)}_{n}\left(x\right)|\leq C^{(\lambda)}_{n}\left(1\right)=\frac{{% \left(2\lambda\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\lambda>0$.
 18.14.5 $|C^{(\lambda)}_{2m}\left(x\right)|\leq|C^{(\lambda)}_{2m}\left(0\right)|=\left% |\frac{{\left(\lambda\right)_{m}}}{m!}\right|,$ $-1\leq x\leq 1$, $-\tfrac{1}{2}<\lambda<0$,
 18.14.6 $|C^{(\lambda)}_{2m+1}\left(x\right)|<\frac{-2{\left(\lambda\right)_{m+1}}}{% \left((2m+1)(2\lambda+2m+1)\right)^{\frac{1}{2}}m!},$ $-1\leq x\leq 1$, $-\tfrac{1}{2}<\lambda<0$.
 18.14.7 ${(n+\lambda)^{1-\lambda}(1-x^{2})^{\frac{1}{2}\lambda}|C^{(\lambda)}_{n}\left(% x\right)|<\frac{2^{1-\lambda}}{\Gamma\left(\lambda\right)}},$ $-1\leq x\leq 1$, $0<\lambda<1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

### Laguerre

 18.14.8 ${\mathrm{e}}^{-\frac{1}{2}x}\left|L^{(\alpha)}_{n}\left(x\right)\right|\leq L^% {(\alpha)}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!},$ $0\leq x<\infty$, $\alpha\geq 0$. ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Proved: Koornwinder (1977, Remark 4.1)(proved) Proof sketch: The case $\alpha=0$ follows from the result on monotonic weight functions in §18.2(xii). A&S Ref: 22.14.13 Referenced by: §18.14(i), §18.2(xii) Permalink: http://dlmf.nist.gov/18.14.E8 Encodings: TeX, pMML, png See also: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

### Hermite

 18.14.9 $\frac{1}{(2^{n}n!)^{\frac{1}{2}}}{\mathrm{e}}^{-\frac{1}{2}x^{2}}|H_{n}\left(x% \right)|\leq 1,$ $-\infty. ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: 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 See also: Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

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

## §18.14(ii) Turán-Type Inequalities

### Legendre

 18.14.10 $(P_{n}\left(x\right))^{2}\geq P_{n-1}\left(x\right)P_{n+1}\left(x\right),$ $-1\leq x\leq 1$. ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

### Jacobi

Let $R_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)/P^{(\alpha,\beta)}_{n}\left(1\right)$. 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), §18.14(ii) Permalink: http://dlmf.nist.gov/18.14.E11 Encodings: TeX, pMML, png See also: Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

See Nikolov and Pillwein (2015) for a variant of (18.14.11) when $\alpha=\beta\in(-1,0]$.

### Laguerre

 18.14.12 $(L^{(\alpha)}_{n}\left(x\right))^{2}\geq L^{(\alpha)}_{n-1}\left(x\right)L^{(% \alpha)}_{n+1}\left(x\right),$ $0\leq x<\infty$, $\alpha\geq 0$. ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

### Hermite

 18.14.13 $(H_{n}\left(x\right))^{2}\geq H_{n-1}\left(x\right)H_{n+1}\left(x\right),$ $-\infty. ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(ii), §18.14(ii), §18.14 and Ch.18

## §18.14(iii) Local Maxima and Minima

### Jacobi

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

 18.14.14 $-1=x_{n,0} ⓘ 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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

Then

 18.14.16 $\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|$ $\displaystyle>|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|>\cdots>|P^{(\alpha,% \beta)}_{n}\left(x_{n,m}\right)|,$ $\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|$ $\displaystyle>|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|>\cdots>|P^{(% \alpha,\beta)}_{n}\left(x_{n,m+1}\right)|,$ $\alpha>-\tfrac{1}{2},\beta>-\tfrac{1}{2}.$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18
 18.14.17 $\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|$ $\displaystyle<|P^{(\alpha,\beta)}_{n}\left(x_{n,1}\right)|<\cdots<|P^{(\alpha,% \beta)}_{n}\left(x_{n,m}\right)|,$ $\displaystyle|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|$ $\displaystyle<|P^{(\alpha,\beta)}_{n}\left(x_{n,n-1}\right)|<\cdots<|P^{(% \alpha,\beta)}_{n}\left(x_{n,m+1}\right)|,$ $-1<\alpha<-\tfrac{1}{2},-1<\beta<-\tfrac{1}{2}$.

Also,

 18.14.18 $|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|<|P^{(\alpha,\beta)}_{n}\left(x_{n% ,1}\right)|<\cdots<|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|,$ $\alpha\geq-\tfrac{1}{2}$, $-1<\beta\leq-\tfrac{1}{2}$, ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.14.E18 Encodings: TeX, pMML, png See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18
 18.14.19 $|P^{(\alpha,\beta)}_{n}\left(x_{n,0}\right)|>|P^{(\alpha,\beta)}_{n}\left(x_{n% ,1}\right)|>\cdots>|P^{(\alpha,\beta)}_{n}\left(x_{n,n}\right)|,$ $\beta\geq-\frac{1}{2}$, $-1<\alpha\leq-\tfrac{1}{2}$, ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

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

### Szegő–Szász Inequality

 18.14.20 $\left|\frac{P^{(\alpha,\beta)}_{n}\left(x_{n,n-m}\right)}{P^{(\alpha,\beta)}_{% n}\left(1\right)}\right|>\left|\frac{P^{(\alpha,\beta)}_{n+1}\left(x_{n+1,n-m+% 1}\right)}{P^{(\alpha,\beta)}_{n+1}\left(1\right)}\right|,$ $\alpha=\beta>-\tfrac{1}{2}$, $m=1,2,\dots,n$. ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.14(iii), §18.14(iii), §18.39(v) Permalink: http://dlmf.nist.gov/18.14.E20 Encodings: TeX, pMML, png See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

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

### Laguerre

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

 18.14.21 $0=x_{n,0} ⓘ 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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

Then

 18.14.23 $\displaystyle|L^{(\alpha)}_{n}\left(x_{n,0}\right)|$ $\displaystyle>|L^{(\alpha)}_{n}\left(x_{n,1}\right)|>\cdots>|L^{(\alpha)}_{n}% \left(x_{n,m}\right)|,$ $\displaystyle|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|$ $\displaystyle>|L^{(\alpha)}_{n}\left(x_{n,n-2}\right)|>\cdots>|L^{(\alpha)}_{n% }\left(x_{n,m+1}\right)|.$

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

 18.14.24 $|L^{(\alpha)}_{n}\left(x_{n,0}\right)|<|L^{(\alpha)}_{n}\left(x_{n,1}\right)|<% \cdots<|L^{(\alpha)}_{n}\left(x_{n,n-1}\right)|.$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 See also: Annotations for §18.14(iii), §18.14(iii), §18.14 and Ch.18

### Hermite

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

## §18.14(iv) Positive Sums

### Jacobi

 18.14.25 $\sum_{m=0}^{n}\frac{{\left(\lambda+1\right)_{n-m}}}{(n-m)!}\frac{{\left(% \lambda+1\right)_{m}}}{m!}\mbox{\displaystyle\frac{P^{(\alpha,\beta)}_{m}% \left(x\right)}{P^{(\beta,\alpha)}_{m}\left(1\right)}\geq 0},$ $x\geq-1$, $\alpha+\beta\geq\lambda\geq 0$, $\beta\geq-\tfrac{1}{2}$, $n=0,1,\dots$. ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proved: Gasper (1977, Theorem 5)(proved) Source: Askey and Gasper (1976, Conjecture 1); conjectured Referenced by: Erratum (V1.2.0) Section 18.14 Permalink: http://dlmf.nist.gov/18.14.E25 Encodings: TeX, pMML, png See also: Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18
 18.14.26 $\sum_{m=0}^{n}\frac{P^{(\alpha,\beta)}_{m}\left(x\right)}{P^{(\beta,\alpha)}_{% m}\left(1\right)}\geq 0,$ $x\geq-1$, $n=0,1,\dots$, ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proved: Gasper (1977, Theorem 4); also proved there for $\alpha+\beta\geq-2$, $\beta\geq 0$(proved) Source: Askey and Gasper (1976, (1.7), (1.12)); conjectured Referenced by: §18.14(iv), §18.38(ii) Permalink: http://dlmf.nist.gov/18.14.E26 Encodings: TeX, pMML, png See also: Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18

for $\alpha+\beta\geq 0$, $\beta\geq-\tfrac{1}{2}$ or $\alpha+\beta\geq-2$, $\beta\geq 0$. The case $\beta=0$ of (18.14.26) is the Askey–Gasper inequality (18.38.3).

### Laguerre

 18.14.27 $\sum_{m=0}^{n}\frac{{\left(\lambda+1\right)_{n-m}}}{(n-m)!}\frac{{\left(% \lambda+1\right)_{m}}}{m!}\mbox{\displaystyle\frac{(-1)^{m}L^{(\beta)}_{m}% \left(x\right)}{L^{(\beta)}_{m}\left(0\right)}\geq 0},$ $x\geq 0$,  $\beta,\lambda\geq-\tfrac{1}{2}$,  $n=0,1,\dots$. ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proved: Gasper (1977, Theorem 1)(proved) Source: Askey and Gasper (1976, Conjecture 2); conjectured Referenced by: Erratum (V1.2.0) Section 18.14 Permalink: http://dlmf.nist.gov/18.14.E27 Encodings: TeX, pMML, png See also: Annotations for §18.14(iv), §18.14(iv), §18.14 and Ch.18