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inequalities245 - NOTES ON INEQUALITIES FELIX LAZEBNIK...

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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 Arithmetic-Geometric 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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2 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 non-negative , and if x 0 or, equivalently, 0 x , we say that a is non-positive . 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 non-negative (non-positive) numbers is non- negative (non-positive), 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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