Inequalities

Frontmatter

Chapter 1. Basic (Elementary) Inequalities and Their Application

Abstract

There are many trivial facts which are the basis for proving inequalities. Some of them are as follows:

1.

If x≥y and y≥z then x≥z, for any x,y,z∈ℝ.

 

2.

If x≥y and a≥b then x+a≥y+b, for any x,y,a,b∈ℝ.

 

3.

If x≥y then x+z≥y+z, for any x,y,z∈ℝ.

 

4.

If x≥y and a≥b then xa≥yb, for any x,y∈ℝ⁺ or a,b∈ℝ⁺.

 

5.

If x∈ℝ then x ²≥0, with equality if and only if x=0. More generally, for A _i ∈ℝ⁺ and x _i ∈ℝ,i=1,2,…,n holds \(A_{1} x_{1}^{2} + A_{2} x_{2}^{2} +\cdots+ A_{n} x_{n}^{2} \geq 0\), with equality if and only if x ₁=x ₂=⋯=x _n =0.

 

These properties are obvious and simple, but are a powerful tool in proving inequalities, particularly Property 5, which can be used in many cases.

Zdravko Cvetkovski

Chapter 2. Inequalities Between Means (with Two and Three Variables)

Abstract

In this section, we’ll first mention and give a proof of inequalities between means, which are of particular importance for a full upgrade of the student in solving tasks in this area. It ought to be mentioned that in this section we will discuss the case that treats two or three variables, while the general case will be considered later in Chap. 5.

Zdravko Cvetkovski

Chapter 3. Geometric (Triangle) Inequalities

Abstract

These inequalities in most cases have as variables the lengths of the sides of a given triangle; there are also inequalities in which appear other elements of the triangle, such as lengths of heights, lengths of medians, lengths of the bisectors, angles, etc.

Zdravko Cvetkovski

Chapter 4. Bernoulli’s Inequality, the Cauchy–Schwarz Inequality, Chebishev’s Inequality, Surányi’s Inequality

Abstract

These inequalities fill that part of the knowledge of students necessary for proving more complicated, characteristic inequalities such as mathematical inequalities containing more variables, and inequalities which are difficult to prove with already adopted elementary inequalities. These inequalities are often used for proving different inequalities for mathematical competitions.

Zdravko Cvetkovski

Chapter 5. Inequalities Between Means (General Case)

Abstract

In Chap. 2 we discussed mean inequalities of two and three variables. In this section we will develop their generalization, i.e. we’ll present an analogous theorem for an arbitrary number of variables.

Zdravko Cvetkovski

Chapter 6. The Rearrangement Inequality

Abstract

In this section we will introduce one really useful inequality called the rearrangement inequality. This inequality has a very broad and easy use in proving other inequalities.

Zdravko Cvetkovski

Chapter 7. Convexity, Jensen’s Inequality

Abstract

The main purpose of this section is to acquaint the reader with one of the most important theorems, that is widely used in proving inequalities, Jensen’s inequality. This is an inequality regarding so-called convex functions, so firstly we will give some definitions and theorems whose proofs are subject to mathematical analysis, and therefore we’ll present them here without proof.

Zdravko Cvetkovski

Chapter 8. Trigonometric Substitutions and Their Application for Proving Algebraic Inequalities

Abstract

Very often, for proving a given algebraic inequality we can use trigonometric substitutions that work amazingly well, and can almost always lead the solver to a solution.

Zdravko Cvetkovski

Chapter 9. Hölder’s Inequality, Minkowski’s Inequality and Their Variants

Abstract

In this chapter we’ll introduce two very useful inequalities with broad practical usage: Hölder’s inequality and Minkowski’s inequality. We’ll also present few variants of these inequalities. For that purpose we will firstly introduce the following theorem.

Zdravko Cvetkovski

Chapter 10. Generalizations of the Cauchy–Schwarz Inequality, Chebishev’s Inequality and the Mean Inequalities

Abstract

In Chap. 4 we presented the Cauchy–Schwarz inequality, Chebishev’s inequality and the mean inequalities. In this section we will give their generalizations. The proof of first theorem is left to the reader, since it is similar to the proof of Cauchy–Schwarz inequality.

Zdravko Cvetkovski

Chapter 11. Newton’s Inequality, Maclaurin’s Inequality

Abstract

Let a ₁,a ₂,…,a _n be arbitrary real numbers.

Consider the polynomial

$$P(x) = (x + a_{1})(x + a_{2}) \cdots(x + a_{n}) = c_{0}x^{n} +c_{1}x^{n - 1} + \cdots + c_{n - 1}x + c_{n}.$$

Then the coefficients c ₀,c ₁,…,c _n can be expressed as functions of a ₁,a ₂,…,a _n , i.e. we have

For each k=1,2,…,n we define \(p_{k} = \frac{c_{k}}{\binom{n}{k}} = \frac{k!(n - k)!}{n!}c_{k}\).

Zdravko Cvetkovski

Chapter 12. Schur’s Inequality, Muirhead’s Inequality and Karamata’s Inequality

Abstract

In this chapter we will present three very important theorems, which have broad usage in solving symmetric inequalities. In that way we’ll start with following definition.

Zdravko Cvetkovski

Chapter 13. Two Theorems from Differential Calculus, and Their Applications for Proving Inequalities

Abstract

In this section we’ll give two theorems (without proof), whose origins are part of differential calculus, and which are widely used in proving certain inequalities. We assume that the reader has basic knowledge of differential calculus.

Zdravko Cvetkovski

Chapter 14. One Method of Proving Symmetric Inequalities with Three Variables

Abstract

In this section we’ll give a wonderful method that will be used in proving symmetrical inequalities with three variables. I must emphasize that this method is a powerful instrument which can be used for proving inequalities of varying difficulty which can’t be proved with previous methods and techniques. Also I must say that I respect this method so much, because it can be very valuable and workable for all symmetric inequalities.

Zdravko Cvetkovski

Chapter 15. Method for Proving Symmetric Inequalities with Three Variables Defined on the Set of Real Numbers

Abstract

This section will consider one method that is similar to the previous method of Chap. 14, for proving symmetrical inequalities with three variables that will be solvable only by elementary transformations and without major knowledge of inequalities (in the sense that for some of them the student has no need to know the powerful Cauchy–Schwarz, Chebishev, Minkowski and Hölder inequalities).

Zdravko Cvetkovski

Chapter 16. Abstract Concreteness Method (ABC Method)

Abstract

In this section we will present three theorems without proofs (the proofs can be found in T. Puong (Diamonds in Mathematical Inequalities, 2007) which are the basis of a very useful method, the Abstract Concreteness Method (ABC method).

For this purpose we’ll consider the function f(abc,ab+bc+ca,a+b+c), as a one-variable function with variable abc on ℝ, i.e. on ℝ⁺.

Zdravko Cvetkovski

Chapter 17. Sum of Squares (SOS Method)

Abstract

One of the basic procedures for proving inequalities is to rewrite them as a sum of squares (SOS) and then, according to the most elementary property that the square of a real number is non-negative, to prove a certain inequality. This property is the basis of the SOS method.

Zdravko Cvetkovski

Chapter 18. Strong Mixing Variables Method (SMV Theorem)

Abstract

This method is very useful in proving symmetric inequalities with more than two variables. The SMV method (strong mixing variables method) is a simple and concise method that “works” in proving inequalities that have either a too complicated or a too long proof. In order to better describe the given method, first we will give a lemma (without proof) and then we will introduce the reader to the SMV theorem and its applications through exercises. We should point out that this theorem is part of a more comprehensive method, the Mixing Variable method (MV method), which can be found in Puong (Diamonds in Mathematical Inequalities, 2007).

Zdravko Cvetkovski

Chapter 19. Method of Lagrange Multipliers

Abstract

This method is intended for conditional inequalities. It requires elementary skills of differential calculus but it is very easy to apply. We’ll give the main theorem, without proof, and we’ll introduce some exercises to see how this method works.

Zdravko Cvetkovski

Chapter 20. Problems

Abstract

20.1 Let n be a positive integer. Prove that

$$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{n^{2}} < 2.$$

Zdravko Cvetkovski

Chapter 21. Solutions

Abstract

1 Let n be a positive integer. Prove that

$$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots + \frac{1}{n^{2}} < 2.$$

Solution For each k≥2 we have

$$\frac{1}{k^{2}} < \frac{1}{k(k - 1)} = \frac{1}{k - 1} - \frac{1}{k}.$$

So

Zdravko Cvetkovski

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