BSc Students‎ > ‎Elementary Algebra‎ > ‎

### 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 01 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 a1≤a2≤...≤an and b1≤b2≤...≤bn, then n(a1b1 + a2b2 +... + anbn)≥ (a1 + a2 + ...+ an)(b1 + b2 +... + bn)  a1≥a2≥...≥an and b1≤b2≤...≤bn, then n(a1b1 + a2b2 +... + anbn)≤ (a1 + a2 + ...+ an)(b1 + b2 +... + bn)