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1 Algebraic and Analytic MethodsTopics of Discussion

§1.7 Inequalities

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http://dlmf.nist.gov/1.7
See also:
Annotations for Ch.1
Contents
  1. §1.7(i) Finite Sums
  2. §1.7(ii) Integrals
  3. §1.7(iii) Means
  4. §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
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Annotations for §1.7 and Ch.1

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).
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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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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

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}.
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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
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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

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}.
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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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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

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.
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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
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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

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}.
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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:
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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

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}.
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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:
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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,
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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:
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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}),
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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:
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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,
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Symbols:
M(\NVar{r}): weighted mean
A&S Ref:
3.2.4
Permalink:
http://dlmf.nist.gov/1.7.E9
Encodings:
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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,
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Symbols:
\,\mathrm{d}\NVar{x}: differential of x, \int: integral and \phi(x): function
Permalink:
http://dlmf.nist.gov/1.7.E10
Encodings:
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See also:
Annotations for §1.7(iv), §1.7 and Ch.1

For \exp and \ln see §4.2.

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