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Inequalities

  • If A is the arithmetic mean of a set of numbers, G is the geometric mean and H is the harmonic mean then A ≥ G ≥ H

  • If x1,x2,...xn are n positive real numbers such that xl + x2 +. . . . .+ xn is a constant, then their arithmetic mean attains its lowest value and their geometric mean attains its maximum value when xl = x2 = . . . . . = xn = A = G

  • If x1 + x2 + x3 + ...+ xn = c , a constant
                                   n
    for 0<m<1 the maximum value of ∑xim is n1-mcm
                                  i=1
                                   n
    for m<0 and m>1 the minimum value of ∑xim is n1-mcm
                                  i=1
    these values are attained when xl = x2 = . . . . . = xn.

  • If x1,x2,... xn are n positive real numbers
    then |x1 + x2 + x3 + ...+ xn|≤|x1| + |x2| + |x3| + ...+ |xn|
    equality holds only when all the xi have the same sign.

  • Cauchy Schwarz Inequality
    If ai,bi ∈R then
    (a1b1 + a2b2 + ... + anbn)2 ≤ (a12 + a22 + ... + an2)(b12 + b22 + ... + bn2)
    with equality iff ai = cbi ∀ i = 1 to n, where c is a constant.

  • If all the roots of real polynomial equation xn + a1xn-1 + a2xn-2 + ... + an = 0 are real then they lie between
    -a1/n - {(n-1)√(a12 - (2n /(n-1))a2}/n and -a1/n + {(n-1)√(a12 - (2n /(n-1))a2}/n.

  • Weierstrass' Inequality
    If x1,x2,...xn are n positive real numbers less than 1 and sn = x1+x2+...+xn
    Then,
    1-sn ≤ (1 - x1)(1 - x2)...(1 - xn)≤ 1/(1+sn)
    1+sn ≤ (1 + x1)(1 + x2)...(1 + xn)≤ 1/(1-sn) , where it is assumed that sn<1

  • Chebyshev's Inequality
      If a1,a2,...an , b1,b2,...bn ∈ R
      such that
    1. a1≤a2≤...≤an and b1≤b2≤...≤bn, then
      n(a1b1 + a2b2 +... + anbn)≥ (a1 + a2 + ...+ an)(b1 + b2 +... + bn)
    2.  a1≥a2≥...≥an and b1≤b2≤...≤bn, then
      n(a1b1 + a2b2 +... + anbn)≤ (a1 + a2 + ...+ an)(b1 + b2 +... + bn)



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