NOTES ON INEQUALITIES
FELIX LAZEBNIK
Order and inequalities are fundamental notions of modern mathematics. Calcu
lus and Analysis depend heavily on them, and properties of inequalities provide the
main tool for developing these subjects. Often students are not sure what they are
allowed to take for granted when they argue about inequalities. These brief notes
are intended to help them to review the basics, improve their skills in working with
inequalities, and present two inequalities which have many applications.
Here is
the list of sections.
Content:
1. Order on
R
and Basic Properties of Inequalities
2. Solving Inequalities: Case analysis
3. Solving Inequalities: Method of Intervals
4. Proving Inequalities by Induction
5. Jensen’s Inequality
6. The ArithmeticGeometric Mean Inequality (AGM)
7. Problems
8. Hints and Answers to Problems
1.
Order on
R
and Basic Properties of Inequalities
We assume that the reader is well familiar with real numbers (or just reals), with
the algebraic properties of operations on them, and with basic properties of their
ordering. We denote the set of all real numbers by
R
, the set of all negative reals
by
R
−
, and the set of positive reals by
R
+
. Then
R
−
,
{
0
}
, and
R
+
partition
R
,
which is another way of saying that every real number is either negative, or zero,
or positive, and no real number has two of these properties. We take for granted
that the following properties hold.
•
the sum of any two positive reals is positive
•
the product of two positive reals is positive
•
the sum of any two negative reals is negative
•
the product of any two negative reals is positive
•
the product of a positive real and a negative real is negative
•
number
1
is positive
•
x
positive (negative) if and only if
x
−
1
is positive (negative)
•
the ratio of a positive and a negative numbers is negative
Date
: December 30, 2009.
1
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FELIX LAZEBNIK
•
for any two reals
x, y
,
xy
= 0
if and only if
x
= 0
or
y
= 0
•
for any two reals
x, y
,
x < y
if and only if
y
−
x
is positive.
•
for any positive (negative) real
x
, 0
< x
(
x <
0).
For convenience, we introduce another symbol
>
, called
greater
, and we write
x > y
if
y < x.
The abbreviation for (
x < y
)
∨
(
x
=
y
) is
x
≤
y
. If
x
≤
y
, we say that
x
is
at
most
y
, or, equivalently,
y
is
at least
x
. Similarly, for
x
≥
y
. Hence, 3
≤
3 and
3
≤
5 are true. If 0
≤
x
or, equivalently,
x
≥
0, we say that
x
is
nonnegative
,
and if
x
≤
0 or, equivalently, 0
≥
x
, we say that
a
is
nonpositive
.
The following property is used often, and we wish to state it separately.
•
for every
x
,
x
2
≥
0
, and
x
2
= 0
if and only if
x
= 0
.
Moreover, the sum of
n
≥
2
nonnegative (nonpositive) numbers is non
negative (nonpositive), and it is equal to
0
if and only if all addends are
equal to
0
.
Now we want to (finally) prove some more subtle properties which relate the
inequalities on
R
and operations on
R
. (So, please forget that you are familiar with
them!)
Theorem 1.1.
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 Fall '10
 CIOABA
 Calculus, Inequalities, Mathematical Induction, Negative and nonnegative numbers, FELIX LAZEBNIK

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