Page 75 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes p p
or |f + g| p |f||f g| q |g||f g| q
Integrating, we get

p p
p
|f g| dx |f||f g| q |g||f g| dx … (4)
q
Using (3), relation (4) becomes

1 1
p
|f g| dx |f| dx p |g| dx p |f g| dx 1 q
p
p
p
1
p
Dividing each term by |f g| dx q , we get
1 1 q 1 p 1 p
p
p
p
|f g| dx |f| dx |g| dx
1 1 1 1
But 1 1
p q q p

1 1 1
p p p
p
p
p
So |f g| dx |f| dx |g| dx
or f + g f + g
p p p
Hence the proof.

Note Equality hold in Minkowski’s inequality if and only if one of the functions f and g
is a multiple of the other.
Theorem 2: Minkowski’s inequality for 0 < p < 1. If 0 < p < 1 and f, g are non-negative functions
in L , then
p
f + g f + g .
p p p
Proof: For this proceed as in theorem Minkowski’s inequality and applying the Hölder’s
p
q
inequality for 0 < p < 1 for the functions f L and (f + g) p/q L , we get
1 p 1 q
|f||f g| p/q |f| p |f g| p/q q
)
1 p 1 q
|f||f g| p/q |f| p |f g| p … (i)

Also g L , proceeding as above, we get
p
1 p 1 q
p
|g||f g| p/q |g| dx |f g| p … (ii)

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