College of education mathematics department

Application of Minkowski’s inequality

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26 Phan Khánh Vân (Group 1)

5.3.2. Applications in analytics
5.3.3. Applications in algebra

5.3. Application of Minkowski’s inequality

5.3.1. Applications in trigonometry:

Exercise 5.14. Proving inequality:

Solution:
In the Cartesian coordinate system orthogonal, we set:

on the other hand (Consequences of Minkowski's inequality)
Infer
The equal sign occurs if one of the following conditions is satisfied:

Exercise 5.15. Prove that:
Solution:
The given inequality is rewritten in the following form:

In the Cartesian coordinate system, consider the vectors:

Infer:

As a consequence of the Minkowski inequality we infer what we have to prove.

5.3.2. Applications in analytics:

Exercise 5.16. Let x be any positive real number and p, q are positive constants. Find the minimum value of the function:

Solution:
In Cartesian coordinate system perpendicular to Oxy consider three points:
move on ray .
and lies on the bisector of the first angle.
Hence:

We have: (As a consequence of the Minkowski inequality)

So the minimum value of the function is .
Exercise 5.17. Find the maximum value of the sum:

where is the solution of the equationtrong

is a given positive integer, .
Solution:
Substituting
into the given equality, we get:

In the Cartesian coordinate system perpendicular to the vectors:

Hence:

So:

So:

There is equality when n vectors are equal and the angles are chosen such that with i=1,2,….,n. To achieve this, we find . satisfies the following conditions:

So: , where , satisfy the above conditions.
So the maximum value of where is the solution of the given equation is: .

5.3.3. Applications in algebra:

Exercise 5.18. Let x, y, z be different double real numbers. Prove that

Solution:
The inequality to be proved is rewritten in the following form:
On the Cartesian coordinate system, take the points A, B, and C with the following coordinates:

Then (1)
We have (As a consequence of the Minkowski inequality)
The equal sign in (3) occurs vectors are in the same direction, in the same direction, ie:

The system (4), (5) cannot happen because . So in (3) there cannot be an equal sign, i.e . Like that (2), we have something to prove.

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