Classical inequalities Arithmetic mean geometric mean AM GM inequality For any nonnegative numbers a1 an a1 a 2 a n n a1 a2 an n Power mean inequality For positive numbers a1 an and a real number let a 1 a n 1 M a1 an n n a1 an 6 0 0 Then M is an increasing function of unless a1 an in which case M is constant Cauchy s inequality For arbitrary real numbers a1 an b1 bn a1 b1 an bn 2 a21 a2n b21 b2n Furthermore equality holds if and only if the vectors a1 an and b1 bn are proportional Cauchy s inequality is equivalent to the triangle inequality for the 2 norm Triangle inequality For any two vectors x y in IRn kx yk2 kxk2 kyk2 Definition Suppose that f is a real valued function defined on an interval and for any points x y in the interval f x f y x y f 2 2 Then f is convex Convex functions have useful properties Jensen s inequality If w1 wn are positive numbers satisfying w1 wn 1 and x1 xn are any n points in an interval where f is convex then f w1 x1 wn xn w1 f x1 wn f xn Points of maximum If f is convex on a b then the maximum value of f is taken at one of the endpoints i e f x max f a f b Weighted AM GM inequality If x1 xn are nonnegative real numbers and w1 wn are positive numbers satisfying w1 wn 1 then n Y i 1 i xw i 1 n X i 1 w i xi Equality holds if and only if x1 xn Theorem If a and b are nonnegative numbers and p q 1 satisfy 1 p 1 q 1 then a p bq ab p q 1 with equality if and only if ap bq Ho lder s inequality Let x1 xn and y1 yn be nonnegative and let p q 1 satisfy 1 p 1 q 1 Then n X i 1 xi y i n X xpi i 1 1 p n X i 1 yiq 1 q Minkowski s inequality If x1 xn and y1 yn are nonnegative numbers and p 1 then 1 p 1 p 1 p n n n X X X p p p yi xi xi yi i 1 i 1 i 1 Theorem Ho lder Let X xij be an m n matrix with nonnegative elements and let w1 wn be positive numbers satisfying w1 wn 1 Then m Y n X w xijj i 1 j 1 m X n Y xij i 1 j 1 wj Rearrangement inequality Let a1 an b1 bn be two sequences of real numbers and suppose a1 a2 an For each permutation of 1 2 n let n X ak b k k 1 Then is largest when b 1 b n and smallest when b 1 b n Examples 1 Let Hn 1 1 2 n1 Show that n n 1 1 n n Hn for every n IN Hint AM GM inequality 2 Show that if 0 a b c 1 then a b c a 1 b 1 c 1 1 b c 1 c a 1 a b 1 Hint convexity 3 Let n ZZ n 6 0 1 Prove that if sin2n 2 A cos2n 2 A 1 cos2n B sin2n B holds then it holds for all n ZZ Hint inequality 1 2 4 Let a mm 1 nn 1 mm nn where m n IN Prove that am an mm nn Hint Bernoulli s inequality 5 For x y z 0 establish the inequality x x z 2 y y z 2 x z y z x y z and determine when equality holds Hint find and use symmetry 6 For a positive number x and an integer n prove that n X bkxc bnxc k 1 k Hint rearrangement inequality 3
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