Geometric Inequalities

7 Key excerpts on "Geometric Inequalities"

• 

  eBook - ePub

  Expanding Mathematical Toolbox: Interweaving Topics, Problems, and Solutions

  • Boris Pritsker(Author)
  • 2023(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  9 Inequalities Wonderland

  DOI: 10.1201/9781003359500-9

  Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.

  Paul Dirac

  Inequalities are very important and are broadly used in all branches of mathematics. Learning about them paves the way to a deeper understanding of various topics in algebra, calculus, geometry, and number theory.

  Solving problems concerning inequalities involve immensely different techniques than solving equations. This is like operating in a different universe. In most cases, you are not required to find a specific value, but rather the range of values satisfying an inequality. To prove inequality holds true, one needs to determine a technique allowing one to verify the validity of a statement in question satisfying restrictions imposed on variables. We will go over several such techniques and reveal useful tricks in dealing with inequalities proofs. In this chapter, we will also show how inequalities can be applied (sometimes, unexpectedly) to solving many interesting problems that originally have nothing to do with inequalities. We will demonstrate how crucial it is to properly utilize them in revealing valuable connections between math disciplines and taking advantage of those connections for making solutions to complicated problems not just manageable, but efficient and elegant.

  One of the well-known classic inequalities that prove useful whenever we try to compare the numerical expressions or assess their upper or lower bounds is AM-GM Inequality introduced and frequently referred to in previous chapters:

  The arithmetic mean of any n nonnegative real numbers is greater than or equal to their geometric mean. The two means are equal if and only if all the numbers are equal:

  a 1

  +

  a 2

  + … +

  a n

  n

  ≥

  a 1

  ·

  a 2

  · … ·

  a n

  n

  .

  PROBLEM 9.1.

  Which one is the greater of two numbers

  a =

  25 3

  +

  9 3

  or

  b =

  14350 6

  ?

  SOLUTION.

  Applying AM-GM Inequality to evaluate a, gives

  a =

  25 3

  +

  9 3

  ≥ 2 ⋅

  25 ⋅ 9

  6

  =

  64 ⋅ 25 ⋅ 9

  6

  =

  14400 6

  >

  14350 6

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  Inequalities

  Theory of Majorization and Its Applications

  • Ingram Olkin, Albert W. Marshall(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  This unification also has the advantage of suggesting new inequalities. Because of the repeated references made to it, the book Geometric In-equalities, cited above, is referred to as G.I. Other inequalities are from the American Mathematical Monthly, which is referred to more simply as the Monthly. References where the dates appear without parentheses are taken from G.I. and are not repeated in the bibliography of this book. That majorization can play a role in generating inequalities for the tri-angle was noted by Steinig (1965), who obtained majorizations between the sides of a triangle and the exradii, and between the sides of a triangle and the medians. For the triangle, the sum of the angles is fixed, and majoriza-tion arises quite naturally. Many inequalities for the angles of a triangle are obtained as a direct application of a majorization using a Schur-convex function. Subsequent to the preparation of this chapter, a paper by Oppenheim (1971) was published in 1978 that also contains the idea of using majoriza-tion to obtain inequalities for the triangle. 192 Sect. A INEQUALITIES FOR THE ANGLES OF A TRIANGLE 193 In presenting these inequalities, the case of equality (when it can be achieved) is often readily identified. The reason for this is that if φ is strictly Schur-convex, then in an inequality of the form φ(χ) > φ(α) for all x >■ a, or of the form φ(χ) < φ(α) for all x -< a, equality holds only if x is a permutation of a. The Schur-convex functions used in this chapter are strictly Schur-convex (mostly as a consequence of 3.C.l.a). Trigonometric inequalities for the triangle comprise Section A and are organized according to whether they relate to the sines, cosines, or tangents of the angles of a triangle. Other inequalities for the cotangents, secants, and consecants of the angles are obtainable, but are omitted because they follow similar patterns and tend to be repetitive.

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  Selected Topics in Geometry with Classical vs. Computer Proving

  • Pavel Pech(Author)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  This is based on the expression of a given inequality in terms of other geometric magni-tudes from which the inequality is seen. Using this method the inequality of Euler is solved. Besides computer proofs we shall show classical solutions as well, to compare both approaches. 163 164 Selected Topics in Geometry with Classical vs. Computer Proving 7.1 Inequality between the diagonals of an n -gon The best known geometric inequality is probably the triangle inequality which says that the sum of two sides of a triangle is greater than the third side. We will deal with the inequality between the sides and diagonals of an n -gon. To simplify the situation we will use the word “diagonal” for any segment joining two vertices of an n -gon. First look at a quadrilateral. We start with an investigation of equalities between diagonals holding for various types of quadrilaterals. 7.1.1 Parallelogram law In this part we will explore the equality between the sum of squares of sides and the sum of squares of diagonals of a parallelogram. This equality is known as the parallelogram law [44]: Theorem 7.1 (Parallelogram law). Given a parallelogram with the side lengths a, b and diagonals e, f . Then 2( a 2 + b 2 ) = e 2 + f 2 . (7.1) Let us prove the relation (7.1). Denote the vertices of a parallelogram by the letters A, B, C, D and its side lengths and diagonals by a = | AB | = | CD | , b = | BC | = | DA | , e = | AC | , f = | BD | . Choose a Cartesian coordinate x y b a e f A=[0,0] B=[a,0] C=[x,y] D=[x-a,y] j Fig. 7.1 Parallelogram law: 2( a 2 + b 2 ) = e 2 + f 2 system such that A = [0 , 0] , B = [ a, 0] , C = [ x, y ] , D = [ x − a, y ] (Fig. 7.1). Then Geometric Inequalities 165 b = | BC | = | AD | ⇔ h 1 : ( x − a ) 2 + y 2 − b 2 = 0 , e = | AC | ⇔ h 2 : x 2 + y 2 − e 2 = 0 , f = | BD | ⇔ h 3 : ( x − 2 a ) 2 + y 2 − f 2 = 0 . The conclusion c of a statement has the form c : 2( a 2 + b 2 ) − ( e 2 + f 2 ) = 0 .

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  College Geometry

  A Unified Development

  • David C. Kay(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  281 6 Inequalities.for. Quadrilaterals:.Unified. Trigonometry In order to advance our knowledge of the two non-Euclidean geometries, it is necessary to develop a non-Euclidean “trigonometry.” This necessi-tates the development of a few inequalities for midlines in right triangles and Lambert quadrilaterals. That will be the goal of the first part of this chapter. The development of unified trigonometry will then occupy the second half—material that includes a special “Pythagorean theorem” for the two non-Euclidean geometries. 6 .1. An.Inequality.Concept.for.Unified.Geometry One of the major, defining concepts for unified geometry involves the angle-sum for triangles. In elliptic geometry, the angle-sum of a triangle is greater than 180, while in hyperbolic geometry, the angle-sum is less than 180 (and in Euclidean geometry, of course, it is 180). It is convenient at this point to introduce a special type of inequality that will apply to all three geometries simultaneously. It might be referred to as a unified inequality . While it does not necessarily simplify proofs (more often than not, proofs must be constructed separately for the three geometries), it does save in writing and in stating the inequalities concisely. The idea is this: We let a * b denote the relation in unified geometry that represents a < b , a = b , or a > b , respectively, in hyperbolic, Euclidean, or elliptic geometry. Instead of the asterisk, a symbol more suggestive of inequality will be used: a “curved” inequality symbol, to distinguish this from ordi-nary inequality. The formal definition is then Definition If a and b are real numbers, write a ≺ b iff a < b in hyperbolic geometry, a = b in Euclidean geometry, or a > b in elliptic geometry. Define a ≻ b iff b ≺ a . It is immediately clear that this new relation (inequality) has all the attributes of ordinary inequality for real numbers, since it coincides with

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  Sources in the Development of Mathematics

  Series and Products from the Fifteenth to the Twenty-first Century

  • Ranjan Roy(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 Inequalities 6.1 Preliminary Remarks In his 1928 presidential address to the London Mathematical Society, G. H. Hardy observed, “A thorough mastery of elementary inequalities is to-day one of the first necessary qualifications for research in the theory of functions.” He also recalled, “I think that it was Harald Bohr who remarked to me that that ‘all analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.’ ” It is surprising, however, that the history of one of our most basic inequalities, the arithmetic and geometric means inequality (AMGM), is tied up with the theory of algebraic equations. Inequalities connected with the symmetric functions of the roots of an equation were used to determine the number of that equation’s positive and negative roots. In 1665–66, Newton laid down the foundation in this area when, in order to determine the bounds on the number of positive and negative roots of equations, he stated a far-reaching generalization of Descartes’s rule of signs. The arithmetic and geometric means inequality states that if there are n nonnega- tive numbers a 1 ,a 2 ,...,a n and there are n positive numbers q 1 ,q 2 ,...,q n , such that ∑ n i =1 q i = 1, then a q 1 1 a q 2 2 ··· a q n n ≤ n  i =1 q i a i , (6.1) where equality holds only when all the a i are equal. The theory of equations has an interesting connection with the case for which q i = 1/n: (a 1 a 2 ··· a n ) 1/n ≤ n  i =1 a i /n. (6.2) Note that when n = 2, the AMGM is simply another form of (a 1 − a 2 ) 2 ≥ 0; (6.3) this case can probably be attributed to Euclid. The nature of the relationship between the AMGM and algebraic equations is clear from (6.2). To see this, suppose that 81 82 Inequalities a 1 ,a 2 ,...,a n are the roots of x n − A 1 x n−1 + A 2 x n−2 + ··· + (−1) n A n = 0. Then (6.2) is identical with the inequality A 1/n n ≤ A 1 .

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  Series and Products in the Development of Mathematics: Volume 1

  • Ranjan Roy(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 Inequalities 6.1 Preliminary Remarks In his 1928 presidential address to the London Mathematical Society, 1 G. H. Hardy observed, “A thorough mastery of elementary inequalities is to-day one of the first necessary qualifications for research in the theory of functions.” He also recalled, “I think that it was Harald Bohr who remarked to me that ‘all analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.’ ” It is surprising, however, that the history of one of the most basic inequalities, the arithmetic and geometric means inequality (AMGM), is tied up with the theory of algebraic equations. Inequalities connected with the symmetric functions of the roots of an equation were used to determine the number of that equation’s positive and negative roots. In 1665–1666, Newton laid down the foundation in this area when, in order to determine the bounds on the number of positive and negative roots of equations, he stated a far-reaching generalization of Descartes’s rule of signs. The arithmetic and geometric means inequality states that if there are n nonnegative numbers a 1 ,a 2 , . . . ,a n and there are n positive numbers q 1 ,q 2 , . . . ,q n , such that ∑ n i =1 q i = 1, then a q 1 1 a q 2 2 · · · a q n n ≤ n  i =1 q i a i , (6.1) where equality holds only when all the a i are equal. The theory of equations has an interesting connection with the case for which q i = 1 n : (a 1 a 2 · · · a n ) 1 n ≤ n  i =1 a i n . (6.2) 1 Hardy (1929). 116 6.1 Preliminary Remarks 117 Note that when n = 2, the AMGM is simply another form of (a 1 − a 2 ) 2 ≥ 0; (6.3) this case can probably be attributed to Euclid. The nature of the relationship between the AMGM and algebraic equations is clear from (6.2). To see this, suppose that a 1 ,a 2 , . . . ,a n are the roots of x n − A 1 x n−1 + A 2 x n−2 + · · · + (−1) n A n = 0. Then (6.2) is identical with the inequality A 1 n n ≤ A 1 n .

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  Interpretation of Algebraic Inequalities

  Practical Engineering Optimisation and Generating New Knowledge

  • Michael T. Todinov, Michael Todinov(Authors)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  2   Basic Algebraic Inequalities DOI: 10.1201/9781003199830-2 2.1 Basic Algebraic Inequalities Used for Proving Other Inequalities 2.1.1 Basic Properties of Algebraic Inequalities and Techniques for Proving Algebraic Inequalities Inequalities are statements about expressions or numbers which involve the symbols ‘ < ’ (less than), ‘ > ’ (greater than), ‘ ≤ ’ (less than or equal to) or ‘ ≥ ’ (greater than or equal to). The basic rules related to handling algebraic inequalities can be summarised as follows: For any real numbers a and b, exactly one of the following holds: a < b, a = b, a > b. If a > b and b > c, then a > c If a > b, adding the same number c to both sides of the inequality does not alter its. direction: a + c > b + c Multiplying both sides of an inequality by (-1) reverses the direction of the inequality: if a > b, then − a < − b ; if a < b, then − a > − b If a > 0 and b > 0, then a b > 0 By using the basic rules, the next basic properties can be established (see Todinov, 2020a for more details and proofs) For any real number x, x 2 ≥ 0. The equality holds if and only if x = 0. If x > y, t > 0, then x t > y t and x / t > y / t. From this property, it follows that if 0 < x < 1, then x 2 < x. If x > y > 0, u > v > 0, then x u > y v and x / v > y / u. From this property, it follows that if a > b > 0,. then a 2 > b 2. If x > 0, y > 0, x ≠ y and x 2 > y 2, then x > y. If a strictly increasing function is applied to both sides of an inequality, the inequality will still hold

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