§1.7 Inequalities
Contents
§1.7(i) Finite Sums
In this subsection and are positive constants.
¶ Cauchy–Schwarz Inequality
Equality holds iff , ; .
Conversely, if for all such that , then .
¶ Hölder’s Inequality
For , , , ,
Equality holds iff , ; .
Conversely, if for all such that , then .
¶ Minkowski’s Inequality
For , , ,
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
Equality holds iff for all .
¶ Hölder’s Inequality
For , , , ,
Equality holds iff for all .
¶ Minkowski’s Inequality
For , , ,
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).
with equality iff .
with equality iff , or and some .
with equality iff , or and some .