Exercise 2.4. Let . Prove the inequality
Solution:
Applying the Cauchy-Schwarz inequality we get
Hence,
So it suffices to show that
which is equivalent to , and clearly holds.
Equality occurs iff .
Exercise 2.5. Let be positive real numbers. Prove the inequality
Solution:
By the Cauchy-Schwarz inequality (Corollary 4.3) we have
Also we have
By (1) and (2) we get
Equality occurs iff .
Exercise Let be real numbers such that . Prove the inequality
Solution:
By the Cauchy-Schwarz inequality we have
i.e.
Furthermore
i.e.
Using (1) and (2) we obtain the required inequality.
Equality occurs iff .
CHAPTER 3: HOLDER’S INEQUALITY, MINKOWSKI’S INEQUALITY 3.1. Holder’s inequality 3.1.1. Young's Inequality
We write again the Bernoulli's inequality
Which is valid for We introduce the following notation:
This assumes that From the condition it also follows that . Substituting this in our inequality, we have:
Multiply both sides by . Consequently,
It implies
We got Young's inequality.
Redesignating , we can write Young's inequality in the following form:
Note that for p > 1, Young’s inequality is written with the opposite sign:
3.1.2. (Holder's inequality)
Let be positive real numbers and be such that
.
Then
Equality occurs if and only if
Proof 1:
By Young's inequality for
we obtain
Adding the inequalities , for , we obtain
i.e.
Obviously equality occurs if and only if
Proof 2:
The function for and is strictly convex, and for is strictly concave (Example 7.2).
Let , then by Jensen's inequality we obtain
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