College of education mathematics department

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26 Phan Khánh Vân (Group 1)

Exercise 2.1. Using Bernoulli's inequality and substitute and to prove

Solution:
Bernoulli's inequality: , where .
Put , then by Bernoulli's inequality, we have

Taking root both side as , we get

Put for , then by Bernoulli's inequality, we have

Reciprocal rule as , we obtain

Taking root both side, we have

This is equivalent to

Therefore,

Exercise 2.2. Using Bernoulli's inequality and substitute and to prove that

Solution:
Bernoulli's inequality: , where
Put , then by Bernoulli's inequality, we have

Taking root both side as , we get

Put for , then by Bernoulli's inequality, we have

This is equivalent to
Reciprocal rule as , , we obtain:

Taking root both side, we have:

Or
Therefore,

Exercise 2.3. Let be positive numbers. Prove that

Solution:
The given inequality is equivalent to

or

By Bernoulli's inequality, we have

From and , we have

or

On the other hand,

Hence

This is equivalent to

2.2. The Cauchy–Schwarz inequality:

2.2.1. Theorem (Cauchy-Schwarz inequality)

Let and , be real numbers. Then we have

i.e.
Equality occurs if and only if the sequences and , ) are proportional, i.e. .

Proof 1:
The given inequality is equivalent to

Let .
If then clearly , and inequality (4. ) is true.
So let us assume that .
Inequality (4.4) is homogenous, so we may normalize with

i.e. we need to prove that

Since (following AM – GM inequality), we have

as required.
Equality occurs if and only if

Proof 2:
Consider the quadratic trinomial

This trinomial is non-negative for all , so its discriminant is not positive, i.e.

as required. Equality holds if and only if , i.e.

Now we'll give several consequences of the Cauchy-Schwarz inequality which have broad use in proving other inequalities.

Let be real numbers and . Then we have

2.2.2. Corollary:

Proof :
(1) The given inequality is equivalent to

which is clearly true.
Equality occurs iff i.e. .
(2) If we apply inequality from the first part twice, we get

Equality occurs iff .
Also as you can imagine there must be some generalization of the previous corollaries. Namely the following result is true.

Let be real numbers such that . Then

with equality if and only if .

2.2.3. Corollary:

Proof :
The proof is a direct consequence of the Cauchy-Schwarz inequality.

2.2.4. Corollary:

Let be real numbers. Then

Proof:
By induction by .
For we have equality.
For we have

which is the Cauchy-Schwarz inequality.
For , let the given inequality hold, i.e.

For we have

So the given inequality holds for every positive integer .
The next result is due to Walter Janous, and is considered by the author to be a very important result, which has broad use in proving inequalities.

2.2.5. Theorem (Chebishev's inequality)

Let and be real numbers. Then we have

i.e.
Equality occurs if and only if or .

Proof:
For all we have

i.e.

By (4.4) we get

Equality holds iff we have equality in (4.3), i.e.
or .
Note Chebishev's inequality is also true in the case when and But if (or the reverse) then we have

2.2.6. Exercises: