SOME INEQUALITIES MIGUEL A LERMA (Last updated: August 12, 2014) Introduction These are a few useful inequalities Most of them are presented in two versions: in sum form and in integral form More generally they can be viewed as inequalities involving vectors, the sum version applies to vectors in Rn and the integral version applies to spaces of functions First a few notations and definitions Absolute value The absolute value of x is represented |x| Norm Boldface letters line u and v represent vectors Their scalar product is represented u·v In Rn the scalar product of u = (a1 , , an ) and v = (b1 , , bn ) is n u·v = b i i=1 For functions f, g : [a, b] → R their scalar product is b f (x)g(x) dx a The p-norm of u is represented u p If u = (a1 , a2 , , an ) ∈ Rn , its p-norm is: n u p |ai |p = 1/p i=1 For functions f : [a, b] → R the p-norm is defined: 1/p b f p p |f (x)| dx = a For p = the norm is called Euclidean MIGUEL A LERMA Convexity A function f : (a, b) → R is said to be convex if f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for every x, y ∈ (a, b), ≤ λ ≤ Graphically, the condition is that for x < t < y the point (t, f (t)) should lie below or on the line connecting the points (x, f (x)) and (y, f (y)) f(y) f(x) f(t) x y t Figure Convex function Inequalities Arithmetic-Geometric Mean Inequality (Consequence of convexity of ex and Jensen’s inequality.) The geometric mean of positive numbers is not greater than their arithmetic mean, i.e., if a1 , a2 , , an > 0, then n 1/n i=1 ≤ n n i=1 Equality happens only for a1 = · · · = an (See also the power means inequality.) Arithmetic-Harmonic Mean Inequality The harmonic mean of positive numbers is not greater than their arithmetic mean, i.e., if a1 , a2 , , an > 0, then ≤ n n i=1 1/ai n n i=1 Equality happens only for a1 = · · · = an This is a particular case of the Power Means Inequality SOME INEQUALITIES 3 Cauchy (Hăolder for p = q = 2.) |u ã v| ≤ u n n b2i a2i ≤ i=1 i=1 i=1 b f (x)g(x) dx v n b i b b |f (x)|2 dx ≤ |g(x)|2 dx a a a Chebyshev Let a1 , a2 , , an and b1 , b2 , , bn be sequences of real numbers which are monotonic in the same direction (we have a1 ≤ a2 ≤ · · · ≤ an and b1 ≤ b2 ≤ · · · ≤ bn , or we could reverse all inequalities.) Then n n n 1 b i ≥ bi n i=1 n i=1 n i=1 Note that LHS − RHS = 2n2 (ai − aj )(bi − bj ) ≥ i,j Geometric-Harmonic Mean Inequality The harmonic mean of positive numbers is not greater than their geometric mean, i.e., if a1 , a2 , , an > 0, then n n n i=1 1/ai 1/n ≤ i=1 Equality happens only for a1 = · · · = an This is a particular case of the Power Means Inequality Hă older If p > and 1/p + 1/q = then |u · v| ≤ u n n p b i ≤ v 1/p q n |bi |p |ai | i=1 i=1 1/p 1/q q |f (x)| dx a b p f (x)g(x) dx ≤ 1/q i=1 b b a p |g(x)| dx a MIGUEL A LERMA Jensen If ϕ is convex on (a, b), x1 , x2 , , xn ∈ (a, b), λi ≥ (i = 1, 2, , n), ni=1 λi = 1, then n n λ i xi ≤ ϕ i=1 λi ϕ(xi ) i=1 MacLaurin’s Inequalities Let ek be the kth degree elementary symmetric polynomial in n variables: xi xi · · · xi k ek (x1 , x2 , , xn ) = 1≤i1 0, then n asi n 1/s i=1 i.e., if s > t > 0, then u s 1/t ati ≤ , i=1 ≤ u t 13 Power Means Inequality Let r be a non-zero real number We define the r-mean or rth power mean of non-negative numbers a1 , , an as follows: r M (a1 , , an ) = n 1/r n ari i=1 If r < 0, and ak = for some k, we define M r (a1 , , an ) = The ordinary arithmetic mean is M , M is the quadratic mean, M −1 is the harmonic mean Furthermore we define the 0-mean to be equal to the geometric mean: n M (a1 , , an ) = 1/n i=1 Then for any real numbers r, s such that r < s, the following inequality holds: M r (a1 , , an ) ≤ M s (a1 , , an ) Equality holds if and only if a1 = · · · = an , or s ≤ and ak = for some k (See weighted power means inequality) 6 MIGUEL A LERMA 14 Power Means Sub/Superadditivity We use the definition of r-mean given in subsection 13 Let a1 , , an , b1 , , bn be non-negative real numbers (1) If r > 1, then the r-mean is subadditive, i.e.: M r (a1 + b1 , , an + bn ) ≤ M r (a1 , , an ) + M r (b1 , , bn ) (2) If r < 1, then the r-mean is superadditive, i.e.: M r (a1 + b1 , , an + bn ) ≥ M r (a1 , , an ) + M r (b1 , , bn ) Equality holds if and only if (a1 , , an ) and (b1 , , bn ) are proportional, or r ≤ and ak = bk = for some k 15 Radon’s Inequality For real numbers p > 0, x1 , , xn ≥ 0, a1 , , an > 0, the following inequality holds: n k=1 ( xp+1 k p ≥ ak ( n p+1 k=1 xk ) n p k=1 ak ) Remark: Radons Inequality follows from Hăolders |u·v| ≤ u p+1 v q , 1/q 1/q 1/q 1/q + 1q = with u = (x1 /a1 , , xn /an ), v = (a1 , , an ), p+1 16 Rearrangement Inequality For every choice of real numbers x1 ≤ · · · ≤ xn and y1 ≤ · · · ≤ yn , and any permutation xσ(1) , , xσ(n) of x1 , , xn , we have xn y1 + · · · + x1 yn ≤ xσ(1) y1 + · · · + xσ(n) yn ≤ x1 y1 + · · · + xn yn If the numbers are different, e.g., x1 < · · · < xn and y1 < · · · < yn , then the lower bound is attained only for the permutation which reverses the order, i.e σ(i) = n − i + 1, and the upper bound is attained only for the identity, i.e σ(i) = i, for i = 1, , n 17 Schur If x, y, x are positive real numbers and k is a real number such that k ≥ 1, then xk (x − y)(x − z) + y k (y − x)(y − z) + z k (z − x)(z − y) ≥ For k = the inequality becomes x3 + y + z + 3xyz ≥ xy(x + y) + yz(y + z) + zx(z + x) SOME INEQUALITIES 18 Schwarz (Hăolder with p = q = 2.) |u · v| ≤ u n n |bi |2 , i=1 f (x)g(x) dx , |ai |2 ≤ i=1 b n b i v i=1 b ≤ a b |g(x)|2 dx |f (x)| dx a a 19 Strong Mixing Variables Method We use the definition of rmean given in subsection 13 Let F : I ⊂ Rn → R be a symmetric, continuous function satisfying the following: for all (x1 , x2 , , xn ) ∈ I such that ≤ x1 ≤ x2 ≤ · · · ≤ xn , F (x1 , x2 , , xn ) ≥ F (t, x2 , , xn−1 , t), where t = M r (x1 , xn ) Then: F (x1 , x2 , , xn ) ≥ F (x, x, , x) , where x = M r (x1 , x2 , , xn ) An analogous result holds replacing ≥ with ≤ 20 Weighted Power Means Inequality Let w1 , , wn be positive real numbers such that w1 + · · · + wn = Let r be a non-zero real number We define the rth weighted power mean of non-negative numbers a1 , , an as follows: 1/r n Mwr (a1 , , an ) wi ari = i=1 As r → the rth weighted power mean tends to: n i aw i Mw0 (a1 , , an ) = i=1 which we call 0th weighted power mean If wi = 1/n we get the ordinary rth power means Then for any real numbers r, s such that r < s, the following inequality holds: Mwr (a1 , , an ) ≤ Mws (a1 , , an ) (If r, s = note convexity of xs/r and recall Jensen’s inequality.) MIGUEL A LERMA References [1] G Hardy, J E Littlewood and G P´olya Inequalities, Second Edition Cambridge University Press, 1952 [2] J Michael Steele The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities Cambridge University Press, 2004 [3] Thomas Foregger, Andrei Ismail, Pedro Sanchez MacLaurin’s inequality (version 4) PlanetMath.org Freely available at http://planetmath.org/MacLaurinsInequality.html