# §1.7 Inequalities

## §1.7(i) Finite Sums

In this subsection and are positive constants.

### ¶ Cauchy–Schwarz Inequality

1.7.1

Equality holds iff , ; .

Conversely, if for all such that , then .

### ¶ Hölder’s Inequality

For , , , ,

1.7.2

Equality holds iff , ; .

Conversely, if for all such that , then .

### ¶ Minkowski’s Inequality

For , , ,

1.7.3

The direction of the inequality is reversed, that is, , when . Equality holds iff , ; .

## §1.7(ii) Integrals

In this subsection and () are real constants that can be , provided that the corresponding integrals converge. Also and are constants that are not simultaneously zero.

### ¶ Cauchy–Schwarz Inequality

1.7.4

Equality holds iff for all .

### ¶ Hölder’s Inequality

For , , , ,

1.7.5

Equality holds iff for all .

### ¶ Minkowski’s Inequality

For , , ,

1.7.6

The direction of the inequality is reversed, that is, , when . Equality holds iff for all .

## §1.7(iii) Means

For the notation, see §1.2(iv).

1.7.7

with equality iff .

1.7.8

with equality iff , or and some .

1.7.9,

with equality iff , or and some .

## §1.7(iv) Jensen’s Inequality

For integrable on , , and convex on 1.4(viii)),

1.7.10

For and see §4.2.