1.3. Theorem:
If are positive numbers, then
with equality only when .
Proof.
Let
According to the lemma we have
i.e.,
Equality holds if only if
, that is .
1.4. Exercises:
Exercise 1.1. Let such that . Prove the inequality
When does equality occur?
Solution:
We have
Since we have
Analogously we get
Adding these three inequalities we obtain
Equality holds if and only if , i.e. . Since we get that equality holds iff .
Exercise 1.2. Let be real numbers. Prove the inequality
When does equality occur?
Solution:
Let .
Then clearly , and it follows that
Similarly as in Exercise 1.1, we can prove that for any
By (2) and (3), we get
Equality occurs iff we have equality in (3), i.e. , from which we deduce that .
CHAPTER 2. BERNOULLI’S INEQUALITY & THE CAUCHY–SCHWARZ
INEQUALITY 2.1. The Inequality and Bernoulli's Inequality 2.1.1. Theorem:
A source for the derivation of many classical inequalities is the simple inequality
where
Proof.
This inequality can be proved using derivatives. Consider the function
provided . Differentiating it, we get
It can be seen that the derivative is zero at . The sign of the derivative to the left and right from the point depends on the value of :
Case 1: If , the derivative changes its sign from plus to minus when passing through the point . In this case we have a maximum at .
Case 2: If or , the derivative changes sign from minus to plus when passing through the point . Therefore, this point is a minimum.
Thus, when , the function in the case 1 decreasing, and in the case 2 it is increasing. Take into account that the function is zero at . Then the following inequalities are true provided :
` for ;
for or ;
Or
for ;
for or .
In the first case (when ), the above inequality can be written as
This relationship is used to prove other classical inequalities.
In the second case (when and ), the inequality can be expressed in the form .
In a particular case, assuming that is a natural number, we obtain the wellknown Bernoulli's inequality:
Let , be real numbers with the same sign, greater then . Then we have
2.1.2. Theorem (Bernoulli's inequality):
Proof.
We'll prove the given inequality by induction.
For we have .
Suppose that for , and arbitrary real numbers , with the same signs, inequality (2.1) holds i.e.
Let , and , be arbitrary real numbers with the same signs.
Then, since have the same signs, we have
Hence,
i.e. inequality (2.1) holds for , and we are done.
Let and . Then .
2.1.3. Corollary (Bernoulli's inequality)
Proof.
According to Theorem 2.1, for , we obtain the required result.
2.1.4. Exercises:
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