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Sachin Kaushal, Lovely Professional University Unit 6: Minkowski Inequalities

Unit 6: Minkowski Inequalities Notes

CONTENTS
Objectives
Introduction

6.1 Minkowski Inequalities
6.1.1 Proof of Minkowski Inequality Theorems
6.1.2 Minkowski Inequality in Integral Form

6.2 Summary
6.3 Keywords
6.4 Review Questions
6.5 Further Readings

Objectives

After studying this unit, you will be able to:

p
p
 Define L -space, conjugate numbers and norm of an element of L -space.
 Understand Minkowski inequality.
 Solve problems on Minkowski inequality.
Introduction

p
In mathematical analysis, the Minkowski inequality establishes that the L spaces are normed
p
vector spaces. Let S be a measure space, let 1 p and let f and g be elements of L (s). Then
p
f + g is in L (s), we have the triangle inequality
f + g f + g
p p p
with equality for 1 < p < if and only if f and g are positively linearly dependent, i.e. f = for
g
some 0. In this unit, we shall study Minkowski’s inequality for 1 p < and for 0 < p < 1. We
shall also study almost Minkowski’s inequality in integral form.
6.1 Minkowski Inequalities

Here, the norm is given by:

1/p
p
f = |f| d
p
if p < , or in the case p = by the essential supremum

f = ess sup |f (x)|.
x S
p
The Minkowski inequality is the triangle inequality in L (S). In fact, it is a special case of the
more general fact

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