Digital Library of Mathematical Functions
About the Project
NIST
1 Algebraic and Analytic MethodsAreas

§1.7 Inequalities

Contents

§1.7(i) Finite Sums

Show Annotations
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

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).
Show Annotations
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

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}}.
Show Annotations
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
Encodings:
TeX, pMML, png

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}}.
Show Annotations
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

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

Show Annotations
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

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.
Show Annotations
Symbols:
dx: differential of x and \int: integral
A&S Ref:
3.2.11
Permalink:
http://dlmf.nist.gov/1.7.E4
Encodings:
TeX, pMML, png

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)dx\leq\left(\int_{a}^{b}(f(x))^{p}dx\right)^{{1/p}}\left(% \int_{a}^{b}(g(x))^{q}dx\right)^{{1/q}}.
Show Annotations
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
Encodings:
TeX, pMML, png

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}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}}.
Show Annotations
Symbols:
dx: 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

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

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

For the notation, see §1.2(iv).

1.7.7H\leq G\leq A,
Show Annotations
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

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}),
Show Annotations
Symbols:
M: weighted mean and n: nonnegative integer
A&S Ref:
3.2.2
Permalink:
http://dlmf.nist.gov/1.7.E8
Encodings:
TeX, pMML, png

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

1.7.9M(r)\leq M(s),r<s,
Show Annotations
Symbols:
M: weighted mean
A&S Ref:
3.2.4
Permalink:
http://dlmf.nist.gov/1.7.E9
Encodings:
TeX, pMML, png

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

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

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)dx\right)\leq\int^{1}_{0}\phi(f(x))dx,

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available.