§1.7 Inequalities

§1.7(i) Finite Sums

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

Cauchy–Schwarz Inequality

 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)

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

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)

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

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)

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

§1.7(ii) Integrals

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

 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)

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

Hölder’s Inequality

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)

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

Minkowski’s Inequality

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)

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

§1.7(iii) Means

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)

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 $M(r)$: 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)

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, Symbols: $M(r)$: 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)

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

For $f$ integrable on $[0,1]$, $a, 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)
 1.7.11 $\mathop{\exp\/}\nolimits\!\left(\int^{1}_{0}\mathop{\ln\/}\nolimits\!\left(f(x% )\right)\mathrm{d}x\right)<\int^{1}_{0}f(x)\mathrm{d}x.$

For $\mathop{\exp\/}\nolimits$ and $\mathop{\ln\/}\nolimits$ see §4.2.