Inequalities



Inequalities

An inequality is defined as “a statement of relative size or order of two objects” [1]. Inequalities arose from the question what occurs when the two objects or equations are not equal to each other. People use inequalities to measure or compare two objects. Inequalities are applied in mathematics when it is difficult to constrain quantities that are difficult to express in a formula. In inequalities, there exists usually no one solution but a set of solutions. There are basically two domains of inequalities. The first domain of inequalities is connected to the shape of a triangle while the second domain is associated with mean. We will examine the first domain related to the triangle.

Probably discovered centuries before, the person attributed with proving the triangle inequality is Euclid.

[pic]

Euclid of Alexandria as he was known during his time so as not to be confused with Euclid of Megara who was a famous philosopher was “the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements” [2]. Because little is known about Euclid many assumptions are made about his life and work. It is thought that he lived during the lifetime of the first Ptolemy and lived in Alexandria, Egypt. One such assumption made about Euclid’s life is that he must have been a pupil of Plato’s Academy in Athens due to his vast knowledge of Eudoxus’ and Theaetetus’ geometry. Although The Elements is Euclid’s most notable work, other works of Euclid’s include Data, On Divisions, Optics, and Phaenomena. The following works of Euclid’s have all been vanished: Conics, Book of Fallacies, Surface Loci, Porisms, and Elements of Music.

Euclid’s most famous work The Elements consists of thirteen books. The importance of The Elements may be contributed to the fact it is the first book that creates an axiomatic system. The logical reasoning of the mathematics demonstrated in The Elements was the first of its kind and later was extremely influential in other areas such as science. The Elements is regarded as one of the most successful textbooks ever written with the number of editions in print second only to the Bible. Euclid commenced the work with five postulates and definitions. It is thought Euclid used earlier resources when writing The Elements because numerous definitions are introduced which were never applied previously such as oblong, rhomboid, and rhombus. The subject of books one through six is plane geometry. Topics of plane geometry discussed in books one and two are “basic properties of triangles, parallels, parallelograms, rectangles, and squares” [2]. The triangle inequality is discussed and proved in book one. The remaining subject matter of books three through thirteen include properties of circles, number theory, the Euclidean algorithm for computing the greatest common divisor of two numbers, irrational numbers, and three dimensional geometry. One of the greatest attributes of The Elements is how understandable Euclid states the theorems and their proofs.

The triangle inequality is discussed under Proposition 20 in Book 1 of Euclid’s The Elements. The following is a recreation by David Joyce a Professor of Mathematics as Clark University of the triangle inequality as it appears in Euclid’s The Elements [3].

[pic]

In any triangle the sum of any two sides is greater than the remaining one.

Let ABC be a triangle.

We say that in the triangle ABC the sum of any two sides is greater than the remaining one, that is, the sum of BA and AC is greater than BC, the sum of AB and BC is greater than AC, and the sum of BC and CA is greater than AB.

Choose a point D such that DA equals to CA and then draw BD. Join DC.

Since DA equal AC, therefore the angle ADC also equals the angle ACD. Therefore the angle BCD is greater than the angle ADC.

Since DCB is a triangle having the angle BCD greater than the angle BDC, and the side opposite the greater angle is greater, therefore DB is greater than BC.

But DA equals AC, therefore the sum of BA and AC is greater than BC.

Similarly we can prove that the sum of AB and BC is also greater than CA, and the sum of BC and CA is greater than AB.

Therefore in any triangle the sum of any two sides is greater than the remaining one.

There are several proofs of the triangle inequality which we will progress through during this paper. The first one to be discussed is the proof of the triangle inequality for real numbers.

Triangle Inequality for Real Numbers

[pic]

Since [pic], then [pic]

Thus, [pic]

Because[pic], then [pic]

which can be written as [pic]

Thus, [pic]

Again since[pic], then[pic].

This proves the triangle inequality for real numbers.

From real numbers, the natural progression is to move into the proof of the triangle inequality for complex numbers. Below is the proof:

Triangle Inequality for Complex Numbers

The triangle inequality states if z and w are complex numbers then [pic]

[pic]

Because [pic]for all [pic]

[pic]

[pic]

[pic]

[pic]

Since [pic]

Take the square root of both sides yields [pic]

(All of the above information came from Heriott Watt University Scholar Math website [4])

Both of these proofs are very similar. Each proof deals with understanding that the absolute value of a number whether real or complex is greater than or equal to that same number. However, the complex number proof is a little more involved because one must also know and understand the properties of complex numbers and their conjugates.

So far we have discussed with the triangle inequality only in a one dimensional space. Now we will look at the triangle inequality in [pic] space. In [pic]space, the triangle inequality is known as the Cauchy-Schwarz Inequality in[pic]named after two famous mathematicians Augustin Louis Baron de Cauchy and Herman Amandus Schwarz.

Augustin Cauchy (1789 - 1857) was born in Paris, France during the French Revolution.

[pic]

At an early age, Lagrange took an interest in young Cauchy’s mathematical education and urged Cauchy’s father to enroll him in Ecole Centrale du Pantheon to study languages. After Ecole Centrale du Pantheon, Cauchy entered Ecole Polytechnique and then the engineering school Ecole des Ponts et Chaussees. After graduating engineering school, Cauchy began work on the Ourcq Canal project. For Cauchy’s first job in 1810, he worked “on port facilities for Napoleon’s English invasion fleet” in Cherbourg [5]. In 1811, Cauchy proved “that the angles of a convex polyhedron are determined by its faces” [5]. He left his Cherbourg in 1813 for health reasons and returned to Paris where he applied for several positions none of which he was offered until 1815 when he was offered the assistant professorship of analysis at his alma mater Ecole Polytechnique. During Cauchy’s period of unemployment, he devoted himself to his mathematical research. From Ecole Polytechnique, Cauchy took a position at the College de France in 1817. There he “lectured on methods of integration which he had discovered, but not published, earlier” [5]. As well as defining an integral, “Cauchy was the first to make a rigorous study of the conditions of convergence of infinite series” [5]. Next, the deposed king Charles X of France offered Cauchy the position of tutor to his grandson, the duke of Bordeaux. This position allowed Cauchy to travel extensively. In exchange for his services, the Charles X made Cauchy a baron. In 1838 upon returning to Paris, Cauchy reclaimed his position at the Academy, but he was not allowed to teach there because he refused to a take an oath of allegiance. After losing the position of mathematics chair at the College de France, Cauchy’s mathematical output included “work on differential equations and applications to mathematical physics” [5]. Cauchy also wrote a four volume book entitled Exercises d’analyse et de physique mathematique which discussed mathematical astronomy. Once Louis Phillippe was overthrown, Cauchy returned to his position at Ecole Polytechnique. An argument involving “a priority claim regarding a result on inelastic shocks” with Duhamel overshadowed Cauchy’s remaining years [5]. On May 23, 1857, Cauchy died rather calmly with is family by his side. Cauchy wrote 789 mathematical papers and is associated with many items in mathematics including the famous inequality we are studying. Some of these items are the theory of series, calculus with the aid of limits and continuity, and the “Cauchy-Kovalevskaya existence theorem that offers a solution of partial differential equations” [5].

The second mathematician in our inequality is Hermann Schwarz (1843 - 1921).

[pic]

He studied at Gewerbeinstitut, which is later to be known as the Technical University of Berlin, with the goal of obtaining a chemistry degree. Not long after coming to Gewerbeinstitut, Schwarz was convinced to change his program of study to mathematics. He continued to study at Gewerbeinstitut until he attained his doctorate. Schwarz researched minimal service areas which is a typical problem of the calculus of variations. In 1865, he revealed a “minimal surface has a boundary consisting of four edges of a regular tetrahedron,” which is known today as Schwarz minimal surface [6]. Upon obtaining his certificate to teach, Schwarz acquired positions at the University of Halle, Eidgenossische Technische Hochschule, then at Gottingen University, and finally at the University of Berlin in 1892. While at these various institutions, Schwarz researched “the subjects of function theory, differential geometry, and the calculus of variations” [7]. Another area Schwarz researched was conformal mappings. His achievement in this area was connected with the Riemann mapping theorem. Schwarz’s process incorporated the Dirichlet problem which mapped “polygonal regions to a circle” [6]. By performing this method, Schwarz was then able to provide a thorough proof of the Riemann mapping theorem. Schwarz’ most significant work was answering the “question of whether a given minimal surface really yields a minimal area” [6]. Another great achievement of Schwarz’ was the Schwarz function. In this work, Schwarz characterized “a conformal mapping of a triangle with arcs of circles as sides onto the unit disc” [6]. The Schwarz function was one of the first examples of an automorphic function which was a helped Klein and Poincare to originate the theory of automorphic functions. According to O’Connor and Robertson, it is not surprising that Schwarz discovered a “special case of the general result now known as the Cauchy-Schwarz inequality.” A great deal of Schwarz’ work is characterized by analyzing specialized and narrow questions and applying much more general methods for solving them.

A third mathematician’s can be added to the Cauchy-Schwarz inequality. His name is Viktor Bunyakovsky born in the Ukraine in 1804 (1804 – 1889).

[pic]

Bunyakovsky was a pupil of Cauchy’s in Paris, where he remained until 1826. Upon returning to St. Petersburg, Bunyakovsky became a central figure in bringing mathematics to Russia and the Russian Empire. Bunyakovsky brought to Russia the Cauchy’s theories as well as probabilistic theories of the French. Bunyakovsky taught mathematics and mechanics at several institutions in St. Petersburg including the First Cadet Corps, the Communications Academy, and the Naval Academy. From 1846 to 1880, Bunyakovsky held a position at the University of St. Petersburg. Bunyakovsky did most of his research at the St. Petersburg Academy of Sciences. He held the position of vice-president of the Academy of Science until his death. Although not always credited, Bunyakovsky is “best known for his discovery of the Cauchy-Schwarz inequality” some twenty five years prior to Schwarz’ own discovery [8]. Other work of Bunyakovsky includes “new proof of Gauss’ law of quadratic reciprocity” which is an area of number theory, applied mathematics, and geometry [8]. Bunyakovsky’s work Foundations of the mathematical theory of probability is credited with providing the foundation of Russian probabilistic vocabulary. Over his life, Bunyakovsky was published 150 times for his work in mathematics and mechanics.

Before tackling the Cauchy-Schwarz inequality, we need to understand some of the vocabulary involved. The dimension in which the Cauchy-Schwarz inequality is used is [pic] space also known as Euclidean n-space. In [pic] space there are more than three dimensions. According to Anton and Rorres’ Elementary Linear Algebra [9], the dot product in [pic]is referred to the Euclidean inner product in [pic]which is defined for two vectors ([pic]) and [pic] as [pic]. Another term is the Euclidean norm or Euclidean length of a vector u = ([pic]) in [pic] is[pic]. There are some properties of length in [pic]that we should also know. According to Elementary Linear Algebra [9], these properties hold true if u and v are vectors in [pic] and k is any scalar, then:

(a) [pic]

(b) [pic]if and only if u = 0

(c) [pic]

(d) [pic].

The last property should look familiar because it is the triangle inequality. A visualization of the triangle inequality with respect to vector length is:

[pic]

Triangle Inequality for Euclidean n space

A proof of the triangle inequality for Euclidean n space is shown below and it works in generality.

[pic]

Once we have found the norm or lengths of a certain object or vector, we can now define a distance between two objects or vectors as follows. The Euclidean distance between two points, u and v, in [pic]is defined as[pic]. The properties of distance in [pic] are similar to properties of length. These following properties are true if u,v, and w are vectors in [pic]and k is any scalar:

(a) d(u,v)[pic]0

(b) d(u,v) = if and only if u = v

(c) d(u,v) = d(v,u) and

(d) d(u,v)[pic] d(u,w) + d(w,v).

Once again, the last property should look familiar because it is the triangle inequality only this time it is applied to distance. The proof works in complete generality whenever we have a norm with the properties a – d.

We can then define the distance and the triangle inequality will follow:

[pic]

A visualization of this is also similar to the one for length with the exception that the sides of the triangle are segments and not rays.

[pic]

Both [pic] and d(u,v)[pic] d(u,w) + d(w,v) generalize the results of triangle inequality in Euclidean geometry. For length, [pic] simply may be restated that the sum of two sides of the triangle are greater than or equal to the length of the third side. As for distance, d (u,v)[pic] d(u,w) + d(w,v) may be summed up by stating that the shortest distance between two points is a straight line.

Now we are ready for the Cauchy-Schwarz Inequality in [pic]which states that if u = ([pic]) and v= ([pic]) are vectors in[pic], then[pic]. In other words, the absolute value of the inner product of u and v is less than or equal to norm or length of u multiplied by the norm or length of v which is symbolized by[pic]. The proof of the Cauchy-Schwarz Inequality is shown below.

Cauchy-Schwarz Inequality

If [pic] so the two sides are equal to each other.

Assume[pic]. Let [pic] By the positivity axiom, the Euclidean inner product of any vector with itself is always positive.

[pic].

Using substitution, we can say that[pic].

The inequality implies that the quadratic polynomial [pic] has either no real roots or a double root.

Thus, the discriminant must satisfy the condition of[pic].

Expressing the coefficients a, b, and c in terms of vectors u and v gives [pic] which may be simplified to[pic]. This statement may also be rewritten as[pic].

Taking the square root of both sides and using the fact that [pic]yields[pic].

From the Cauchy-Schwarz inequality which deals with vectors of a finite number of components, we can now move into a discussion of a special type of vector space called sequence space. A sequence space is a vector with an infinite number of components which is represented by v = [pic]that has no ending. There are two inequalities which extend the Cauchy-Schwarz inequality into sequence space.

The first of these inequalities is the Hölder inequality named after the German born mathematician, Otto Hölder (1859 – 1937).

[pic]

Hölder’s education includes studying at the polytechnic in Stuttgart and later at the University of Berlin. While at the University of Berlin, Hölder became interested in algebra because of the influence of Kronecker. In 1882, Hölder gave his dissertation on “analytic functions and summation procedures by arithmetic means” at the University of Tubingen [10]. In 1884, Hölder took the position of lecturer at Gottingen. At Gottingen, Hölder discovered the inequality which is now named for him, began his “work on the convergence of the Fourier series,” and developed a fascination for the area of group theory” [10]. Hölder’s lifetime work includes the Jordan-Hölder theorem which deals with the “uniqueness of the factors groups in a composition series,” contributions to group theory, the creation of inner and outer automorphisms.

A proof of Hölder’s inequality is given below as reconstructed by a Professor Gabriel Nagy at Kansas State University who uses the method induction [11].

Hölder’s Inequality states:

Let [pic] be nonnegative numbers. Let p, q > 1 be real numbers with the property[pic]. Then [pic]. Moreover, one has equality only when the sequences [pic]are proportional.

The proof of Hölder’s Inequality is as follows.

The case n = 1 is trivial.

Case n = 2

Assume[pic]. Otherwise everything is trivial. Define the number

[pic].

Notice that[pic]

The proof of why [pic] follows as:

We were given that [pic]are nonnegative numbers. Let us assume that the second positive number[pic] equals 0. Thus the denominator becomes[pic]. So [pic] . However, [pic]must be a positive number making 1 the greatest value r can be. Therefore,[pic].

We also have[pic].

Notice also that, upon dividing by[pic], the desired inequality

[pic]reads[pic].

It is obvious that this is an equality when[pic]. Assume[pic], and set up the function[pic],

We now apply the Lemma of the following:

Let p, q >1 be such that[pic], and let u and v be two nonnegative numbers, at least one being non-zero. Then the function f: [0,1] [pic] Real numbers defined by [pic]has a unique maximum point at [pic]. The maximum value of f is[pic].

The proof of this Lemma is as follows:

If [pic]

Likewise, if[pic]

[pic]

We immediately get

[pic] and the Lemma again follows.

For the remainder of the proof we are going to assume that[pic]. We concentrate on the first assertion. Obviously [pic]is differentiable on (0, 1), so the “candidates” for the maximum points are 0, 1, and the solutions of the equation

[pic].

Let s be defined as in[pic], so under the assumption that[pic], we clearly have 0 ................
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