Exercise 3.1. Let be positive real numbers such that
.
Prove the inequality
Solution:
Let us denote and . Then the condition becomes
i.e.
By Holder's inequality we have
and since we get
i.e.
as required.
Exercise 3.2. Let be positive real numbers such that . Prove the inequality
Solution
Let us denote
and
By Holder's inequality we obtain
It remains to prove that
Since we deduce that
So it follows that
Exercise 3.3. Let be positive real numbers such that . Prove the inequality
Solution:
Let us denote
and
By Holder's inequality we have
From which it follows that .
Equality occurs iff .
3.2. Minkowski's inequality 3.2.1. Theorem (First Minkowski's inequality) :
Let be positive real numbers and . Then
Equality occurs if and only if .
Proof:
For , we choose such that , i.e.
By Holder's inequality, we have:
i.e. we obtain
Equality occurs if and only if .
3.2.2. Theorem (Second Minkowski's inequality) :
Let be positive real numbers and . Then
Equality occurs if and only if .
Proof:
The function for is a strictly convex and for is a strictly concave
By Jensen's inequality for we obtain
i.e.
If we take and for , by the last inequality we obtain
3.2.3. Theorem (Third Minkowski's inequality) :
Let and , be positive real numbers. Then
Equality occurs if and only if .
Proof:
The proof is a direct consequence of Jensen's inequality for the convex function , with .
3.2.4. Exercises:
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