§1.7 Inequalities

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

§1.7(i) Finite Sums

Notes:: See Hardy et al. (1967, pp. 1–32).
Permalink:: http://dlmf.nist.gov/1.7.i

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

¶ Cauchy–Schwarz Inequality

Keywords:: Cauchy–Schwarz inequalities for sums and integrals, inequalities

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
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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

Keywords:: Hölder’s inequalities for sums and integrals, inequalities

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, $p$ : number and $q$ : number
A&S Ref:: 3.2.8
Permalink:: http://dlmf.nist.gov/1.7.E2
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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

Keywords:: Minkowski’s inequalities for sums and series, inequalities

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
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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

Notes:: See Hardy et al. (1967, pp. 130–147).
Permalink:: http://dlmf.nist.gov/1.7.ii

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

Keywords:: Cauchy–Schwarz inequalities for sums and integrals, inequalities

1.7.4 $\left(\int _{a}^{b}f(x)g(x)dx\right)^{2}\leq\int _{a}^{b}(f(x))^{2}dx\int _{a}^{b}(g(x))^{2}dx.$

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
A&S Ref:: 3.2.11
Permalink:: http://dlmf.nist.gov/1.7.E4
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Equality holds iff $Af(x)=Bg(x)$ for all $x$ .

¶ Hölder’s Inequality

Keywords:: Hölder’s inequalities for sums and integrals, inequalities

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

1.7.5 $\int _{a}^{b}f(x)g(x)dx\leq\left(\int _{a}^{b}(f(x))^{p}dx\right)^{{1/p}}\left(\int _{a}^{b}(g(x))^{q}dx\right)^{{1/q}}.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $p$ : number and $q$ : number
A&S Ref:: 3.2.10
Permalink:: http://dlmf.nist.gov/1.7.E5
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Equality holds iff $A(f(x))^{p}=B(g(x))^{q}$ for all $x$ .

¶ Minkowski’s Inequality

Keywords:: Minkowski’s inequalities for sums and series, inequalities

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

1.7.6 $\left(\int _{a}^{b}(f(x)+g(x))^{p}dx\right)^{{1/p}}\leq\left(\int _{a}^{b}(f(x))^{p}dx\right)^{{1/p}}+\left(\int _{a}^{b}(g(x))^{p}dx\right)^{{1/p}}.$