Inequalities

Contents

Chapter 1. Classic Inequalities 11. Jensens inequality 12. AM GM HM 13. The Cauchy-Schwarz Inequality 34. Chebyshevs Inequality 3

Chapter 2. Methods for proving inequalities 51. Any sum of squares is nonnegative 52. Use monotonic functions 53. Study the behavior of a derivable function 54. Change the variables 65. Inequalities quadratic in one of the variables 66. Use convergent sequences 67. Varia 6

1

CHAPTER 1

Classic Inequalities

1. Jensens inequality

Definition 1. A function f : (a, b) R is called convex (concave) if for any x, y (a, b) and any t [0, 1]the following inequality holds

f(tx+ (1 t)y) ()tf(x) + (1 t)f(y)For < we have [, ] = {t + (1 t), t [0, 1]}. Then the function f is convex (concave) if for any two

points x < y, the graph of the function is under (above) the line determined by the points (x, f(x)) and (y, f(y)).A function is convex if and only if f is concave.Proposition 1. If a function f is two times derivable on the interval (a, b), then f is convex (concave) if and

only if f ()0 on (a, b)

Jensens inequality If f : (a, b) R is a convex (concave) function then for any x1, x2, ..., xn (a, b) and anyt1, t2, ..., tn positive numbers with

nk=1

tk = 1 the following inequality holds

f(t1x1 + t2x2 + ...+ tnxn) ()t1f(x1) + t2f(x2) + ...+ tnf(xn)

Homework Prove Jensens inequality in the particular case t1 = t2 = ...tn =1n.

Hint Prove by induction that the statement for n implies the statements for 2n and n 1.Applications

1. If A,B,C are the angles of a triangle then

a) sinA+ sinB + sinC 33

2

b) for A,B,C pi2, cosA+ cosB + cosC 3

2

c) for A,B,C pi2, tanA+ tanB + tanC 33

2. If m,n are positive integers and x1, x2, ..., xn are nonnegative real numbers, then

xm1 + xm2 + ...x

mn

n(x1 + x2 + ...+ xn

n

)m3. If a+ b = 2 then a4 + b4 2

2. AM GM HMLet x1, x2, ..., xn be positive real numbers. We define their

arithmetic mean AM = x1 + x2 + ...+ xnn

geometric mean GM = nx1x2...xn

harmonic mean HM = n1x1

+1x2

+ ...+1xn

1

2 1. CLASSIC INEQUALITIES

The Inequality of Means AM GM HM , and the equality holds if and only if x1 = x2 = ... = xn.

Proof : Using Jensens inequality for the concave function f(x) = lnx and t1 = t2 = ... = tn =1n

we obtain

AM GM . Then taking xk 1xk

for k = 1, 2, ..., n in AM GM we obtain GM HM .

Applications If x1, x2, ..., xn, a, b, c are positive numbers prove the following inequalities. Determine when theequalities hold.

1. (x1 + x2 + ...+ xn)(

1x1

+1x2

+ ...+1xn

) n2

2. If x1x2...xn = 1, then (1 + x1)(1 + x2)...(1 + xn) 2n

3.x1x2

+x2x3

+ ...xn1xn

+xnx1

n

4. (a+ b)(b+ c)(c+ a) 8abc

5.a

b+ c+

b

c+ a+

c

a+ b 3

2

6.ab

a+ b+

bc

b+ c+

ca

c+ a a+ b+ c

2

7.1

a+ b+

1b+ c

+1

c+ a 1

2

(1a+

1b+

1c

)

8. If a, b, c are nonnegative real numbers, then 9abc (a+ b+ c)(ab+ bc+ ca).

9. If a, , b, c are the sides of a triangle, then (a b+ c)(b c+ a)(c a+ b) abc.

10. In A,B,C are the angles (in radians) of a triangle, then 8 cosA cosB cosC 1.

11. Find the minimum value of 2x+ 3y, knowing that x, y are positive numbers such that x2y3 = 1.

12. If a1, a2, ..., an are positive real numbers with S =n

k=1

ak, prove that

a1S a1 +

a2S a2 + ...+

anS an

n

n 1

13. If a, b, c, d are positive real numbers with a+ b+ c+ d = 1, find the minimum value of1a+

1b+

1c+

1d.

14. If a, b, c, x, y, z are positive real numbers, then

3(a+ x)(b+ y)(c+ z) 3

abc+ 3

xyz

4. CHEBYSHEVS INEQUALITY 3

3. The Cauchy-Schwarz Inequality

If a1, a2, ..., an, b1, b2, ..., bn are real numbers, then

(a21 + a22 + ...+ a

2n)(b

21 + b

22 + ...+ b

2n) (a1b1 + a2b2 + ...+ anbn)2

and the inequality holds if and only ifa1b1

=a2b2

= ... =anbn

.

Proof :1: Induction after n.

Proof :2: Immediate consequence of Lagranges identityni=1

a2i

ni=1

b2i (

ni=1

aibi

)2=i

4 1. CLASSIC INEQUALITIES

Applications1. For any real numbers a, b we have (a+ b)(a2 + b2)(a3 + b3) 4(a6 + b6)2. If a, b, c, respectively A,B,C, are the sides, respectively the angles in radians, of a triangle, then

aA+ bB + cCa+ b+ c

pi3

3. Let n > 1 be an integer. If a1, a2, ..., an are positive real numbers then

aa11 aa22 ...a

ann (a1a2...an)

a1+a2+...+ann

4. If a, b, c are positive real numbers, prove that

2(a3 + b3 + c3) a2(b+ c) + b2(c+ a) + c2(a+ b)

CHAPTER 2

Methods for proving inequalities

1. Any sum of squares is nonnegative

This method is based on the following simple remarks:

1. If x is a real number, then x2 0, with equality only for x = 0.2. If x1, x2, ..., xn are real numbers, then x21 + x

22 + ...x

2n 0, with equality only for x1 = x2 = ... = xn = 0.

3. If x1, x2, ..., xn are real numbers, and a1, a2, ..., an are nonnegative real numbers, then a1x21 + a2x22 + ...+

anx22 0, with equality only for a1x1 = a2x2 = ... = anxn = 0.

Problem 1. For any real numbers a, b, c the following inequalities holda) ab+ bc+ ca a2 + b2 + c2b) abc(a+ b+ c) a4 + b4 + c4

Problem 2. If a, b, c are nonnegative real numbers thena) (a2 + b2)c+ (b2 + c2)a+ (c2 + a2)b 6abcb) 2(a3 + b3 + c3) (a+ b)ab+ (b+ c)bc+ (c+ a)caProblem 3. Let a1, a2, ..., an be real numbers. Prove that

ni=1

nj=1

cos(ai aj) 0.

2. Use monotonic functions

Problem 4. Let a, b, c, d be non-negative real numbers. Prove thata+ b+ c+

b+ c+ d+

c+ d+ a+

d+ a+ b 3a+ b+ c+ d

When is the equality attained? Generalization.

Problem 5. If a, b, c are the sides of a triangle and n is a natural number, then

2n+ 1

ab+ c+ na

+b

c+ a+ nb+

c

a+ b+ nc 1 and a be real numbers.a) If a (0, 1), then (1 + x)a 1 + ax.b) If a / (0, 1), then (1 + x)a 1 + ax.Problem 9. Prove that for any natural number n(

1 +1n

)n+1/3< e 0 prove thatx

(x+ y)(x+ z)+

y(y + x)(y + z)

+z

(z + x)(z + y) 3

2.

Problem 11. Let a, b be real numbers such that ab > 0. Prove that

3

a2b2(a+ b)2

4 a

2 + 10ab+ b2

12.

Determine when equality occurs. Hence, or otherwise, prove for all real numbers a, b that

3

a2b2(a+ b)2

4 a

2 + ab+ b2

3.

Determine the cases of equality.

4.2. Trigonometric substitutions

Problem 12. If n N, and |x| 1, then (1 x)n + (1 + x)n 2n.Problem 13. If a, b [1, 1], then 1 a2 +1 b2 4 (a+ b)2.

5. Inequalities quadratic in one of the variables

Problem 14. For any real numbers a, b, c we have ab+ bc+ ca a2 + b2 + c2Problem 15. If a < b < c < d, then (a+ b+ c+ d)2 > 8(ac+ bd).

6. Use convergent sequences

Problem 16. Show that for every positive integer n,(2n 1e

) 2n12

< 1 3 5 (2n 1)