1 Algebraic and Analytic Methods Topics of Discussion 1.6 Vectors and Vector-Valued Functions 1.8 Fourier Series

§1.7 Inequalities

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Permalink:: http://dlmf.nist.gov/1.7
See also:: Annotations for Ch.1

§1.7(i) Finite Sums
§1.7(ii) Integrals
§1.7(iii) Means
§1.7(iv) Jensen’s Inequality

§1.7(i) Finite Sums

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Notes:: See Hardy et al. (1967, pp. 1–32).
Referenced by:: §1.2(v)
Permalink:: http://dlmf.nist.gov/1.7.i
See also:: Annotations for §1.7 and Ch.1

In this subsection $A$ and $B$ are positive constants.

Cauchy–Schwarz Inequality

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Keywords:: Cauchy–Schwarz, Cauchy–Schwarz inequalities for sums and integrals, inequalities, sums and integrals
See also:: Annotations for §1.7(i), §1.7 and Ch.1

1.7.1		$\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2}\leq\left(\sum^{n}_{j=1}a_{j}^{2}% \right)\left(\sum^{n}_{j=1}b_{j}^{2}\right).$
ⓘ Symbols: $j$ : integer and $n$ : nonnegative integer A&S Ref: 3.2.9 Permalink: http://dlmf.nist.gov/1.7.E1 Encodings: TeX, pMML, png See also: Annotations for §1.7(i), §1.7(i), §1.7 and Ch.1

Equality holds iff $a_{j}=cb_{j}$ , $\forall j$ ; $c=\text{ constant}$ .

Conversely, if $\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2}\leq AB$ for all $b_{j}$ such that $\sum^{n}_{j=1}b_{j}^{2}\leq B$ , then $\sum^{n}_{j=1}a_{j}^{2}\leq A$ .

Hölder’s Inequality

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Keywords:: Hölder’s inequalities for sums and integrals, Hölder’s, inequalities, sums and integrals
See also:: Annotations for §1.7(i), §1.7 and Ch.1

For $p>1$ , $\dfrac{1}{p}+\dfrac{1}{q}=1$ , $a_{j}\geq 0$ , $b_{j}\geq 0$ ,

1.7.2		$\sum^{n}_{j=1}a_{j}b_{j}\leq\left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(% \sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}.$
ⓘ Symbols: $j$ : integer, $n$ : nonnegative integer, $q$ : number and $p>1$ A&S Ref: 3.2.8 Permalink: http://dlmf.nist.gov/1.7.E2 Encodings: TeX, pMML, png See also: Annotations for §1.7(i), §1.7(i), §1.7 and Ch.1

Equality holds iff $a_{j}^{p}=cb_{j}^{q}$ , $\forall j$ ; $c=\text{ constant}$ .

Conversely, if $\sum^{n}_{j=1}a_{j}b_{j}\leq A^{1/p}B^{1/q}$ for all $b_{j}$ such that $\sum^{n}_{j=1}b_{j}^{q}\leq B$ , then $\sum^{n}_{j=1}a_{j}^{p}\leq A$ .

Minkowski’s Inequality

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Keywords:: Minkowski’s, Minkowski’s inequalities for sums and series, inequalities, sums and integrals
See also:: Annotations for §1.7(i), §1.7 and Ch.1

For $p>1$ , $a_{j}\geq 0$ , $b_{j}\geq 0$ ,

1.7.3		$\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p}\leq\left(\sum^{n}_{j=1}a_{j% }^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}.$
ⓘ Symbols: $j$ : integer, $n$ : nonnegative integer and $p>1$ A&S Ref: 3.2.12 Permalink: http://dlmf.nist.gov/1.7.E3 Encodings: TeX, pMML, png See also: Annotations for §1.7(i), §1.7(i), §1.7 and Ch.1

The direction of the inequality is reversed, that is, $\geq$ , when $0<p<1$ . Equality holds iff $a_{j}=cb_{j}$ , $\forall j$ ; $c=\text{ constant}$ .

§1.7(ii) Integrals

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Notes:: See Hardy et al. (1967, pp. 130–147).
Permalink:: http://dlmf.nist.gov/1.7.ii
See also:: Annotations for §1.7 and Ch.1

In this subsection $a$ and $b$ ( $>a$ ) are real constants that can be $\mp\infty$ , provided that the corresponding integrals converge. Also $A$ and $B$ are constants that are not simultaneously zero.

Cauchy–Schwarz Inequality

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Keywords:: Cauchy–Schwarz, Cauchy–Schwarz inequalities for sums and integrals, inequalities, sums and integrals
See also:: Annotations for §1.7(ii), §1.7 and Ch.1

1.7.4		$\left(\int_{a}^{b}f(x)g(x)\,\mathrm{d}x\right)^{2}\leq\int_{a}^{b}(f(x))^{2}\,% \mathrm{d}x\int_{a}^{b}(g(x))^{2}\,\mathrm{d}x.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral A&S Ref: 3.2.11 Permalink: http://dlmf.nist.gov/1.7.E4 Encodings: TeX, pMML, png See also: Annotations for §1.7(ii), §1.7(ii), §1.7 and Ch.1

Equality holds iff $Af(x)=Bg(x)$ for all $x$ .

Hölder’s Inequality

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Keywords:: Hölder’s inequalities for sums and integrals, Hölder’s, inequalities, sums and integrals
See also:: Annotations for §1.7(ii), §1.7 and Ch.1

For $p>1$ , $\dfrac{1}{p}+\dfrac{1}{q}=1$ , $f(x)\geq 0$ , $g(x)\geq 0$ ,

1.7.5		$\int_{a}^{b}f(x)g(x)\,\mathrm{d}x\leq\left(\int_{a}^{b}(f(x))^{p}\,\mathrm{d}x% \right)^{1/p}\left(\int_{a}^{b}(g(x))^{q}\,\mathrm{d}x\right)^{1/q}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p>1$ and $q$ : number A&S Ref: 3.2.10 Permalink: http://dlmf.nist.gov/1.7.E5 Encodings: TeX, pMML, png See also: Annotations for §1.7(ii), §1.7(ii), §1.7 and Ch.1

Equality holds iff $A(f(x))^{p}=B(g(x))^{q}$ for all $x$ .

Minkowski’s Inequality

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Keywords:: Minkowski’s, Minkowski’s inequalities for sums and series, inequalities, sums and integrals
See also:: Annotations for §1.7(ii), §1.7 and Ch.1

For $p>1$ , $f(x)\geq 0$ , $g(x)\geq 0$ ,

1.7.6		$\left(\int_{a}^{b}(f(x)+g(x))^{p}\,\mathrm{d}x\right)^{1/p}\leq\left(\int_{a}^% {b}(f(x))^{p}\,\mathrm{d}x\right)^{1/p}+\left(\int_{a}^{b}(g(x))^{p}\,\mathrm{% d}x\right)^{1/p}.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $p>1$ A&S Ref: 3.2.13 Permalink: http://dlmf.nist.gov/1.7.E6 Encodings: TeX, pMML, png See also: Annotations for §1.7(ii), §1.7(ii), §1.7 and Ch.1

The direction of the inequality is reversed, that is, $\geq$ , when $0<p<1$ . Equality holds iff $Af(x)=Bg(x)$ for all $x$ .

§1.7(iii) Means

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Keywords:: arithmetic mean, geometric mean, harmonic mean, inequalities, means
Notes:: See Hardy et al. (1967, pp. 14, 17, 26).
Permalink:: http://dlmf.nist.gov/1.7.iii
See also:: Annotations for §1.7 and Ch.1

For the notation, see §1.2(iv).

1.7.7		$H\leq G\leq A,$
ⓘ Symbols: $A$ : arithmetic mean, $G$ : geometric mean and $H$ : harmonic mean A&S Ref: 3.2.1 Permalink: http://dlmf.nist.gov/1.7.E7 Encodings: TeX, pMML, png See also: Annotations for §1.7(iii), §1.7 and Ch.1

with equality iff $a_{1}=a_{2}=\dots=a_{n}$ .

1.7.8		$\min(a_{1},a_{2},\dots,a_{n})\leq M(r)\leq\max(a_{1},a_{2},\dots,a_{n}),$
ⓘ Symbols: $n$ : nonnegative integer and $M(\NVar{r})$ : weighted mean A&S Ref: 3.2.2 Permalink: http://dlmf.nist.gov/1.7.E8 Encodings: TeX, pMML, png See also: Annotations for §1.7(iii), §1.7 and Ch.1

with equality iff $a_{1}=a_{2}=\dots=a_{n}$ , or $r<0$ and some $a_{j}=0$ .

1.7.9		$M(r)\leq M(s),$
$r<s$ ,
ⓘ Symbols: $M(\NVar{r})$ : weighted mean A&S Ref: 3.2.4 Permalink: http://dlmf.nist.gov/1.7.E9 Encodings: TeX, pMML, png See also: Annotations for §1.7(iii), §1.7 and Ch.1

with equality iff $a_{1}=a_{2}=\dots=a_{n}$ , or $s\leq 0$ and some $a_{j}=0$ .

§1.7(iv) Jensen’s Inequality

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Keywords:: Jensen’s, Jensen’s inequality, Jensen’s inequality for integrals, inequalities, integrals, sums and integrals
Notes:: See Hardy et al. (1967, pp. 132, 151).
Permalink:: http://dlmf.nist.gov/1.7.iv
See also:: Annotations for §1.7 and Ch.1

For $f$ integrable on $[0,1]$ , $a<f(x)<b$ , and $\phi$ convex on $(a,b)$ (§1.4(viii)),

1.7.10		$\phi\left(\int^{1}_{0}f(x)\,\mathrm{d}x\right)\leq\int^{1}_{0}\phi(f(x))\,% \mathrm{d}x,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\phi(x)$ : function Permalink: http://dlmf.nist.gov/1.7.E10 Encodings: TeX, pMML, png See also: Annotations for §1.7(iv), §1.7 and Ch.1

1.7.11		$\exp\left(\int^{1}_{0}\ln\left(f(x)\right)\,\mathrm{d}x\right)<\int^{1}_{0}f(x% )\,\mathrm{d}x.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function Permalink: http://dlmf.nist.gov/1.7.E11 Encodings: TeX, pMML, png See also: Annotations for §1.7(iv), §1.7 and Ch.1

For $\exp$ and $\ln$ see §4.2.

1.6 Vectors and Vector-Valued Functions 1.8 Fourier Series

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§1.7 Inequalities

Contents

§1.7(i) Finite Sums

Cauchy–Schwarz Inequality

Hölder’s Inequality

Minkowski’s Inequality

§1.7(ii) Integrals

Cauchy–Schwarz Inequality

Hölder’s Inequality

Minkowski’s Inequality

§1.7(iii) Means

§1.7(iv) Jensen’s Inequality