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 x_{1},x_{2},...x_{n} are n positive real numbers such that x_{l} + x_{2} +. . . . .+ x_{n} is a constant, then their arithmetic mean attains its lowest value and their geometric mean attains its maximum value when x_{l} = x_{2} = . . . . . = x_{n} = A = G
If x_{1} + x_{2} + x_{3} + ...+ x_{n} = c , a constant
n
for 0<m<1 the maximum value of ∑x_{i}^{m} is n^{1-m}c^{m}
i=1
n
for m<0 and m>1 the minimum value of ∑x_{i}^{m} is n^{1-m}c^{m}
i=1
these values are attained when x_{l} = x_{2} = . . . . . = x_{n}.
If x_{1},x_{2},... x_{n }are n positive real numbers
then |x_{1} + x_{2} + x_{3} + ...+ x_{n}|≤|x_{1}| + |x_{2}| + |x_{3}| + ...+ |x_{n}|
equality holds only when all the x_{i} have the same sign.
Cauchy Schwarz Inequality
If a_{i},b_{i} ∈R then
(a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n})^{2} ≤ (a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2})(b_{1}^{2} + b_{2}^{2} + ... + b_{n}^{2})
with equality iff a_{i} = cb_{i} ∀ i = 1 to n, where c is a constant.
If all the roots of real polynomial equation x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + ... + a_{n} = 0 are real then they lie between
-a_{1}/n - {(n-1)√(a_{1}^{2} - (2n /(n-1))a_{2}}/n and -a_{1}/n + {(n-1)√(a_{1}^{2} - (2n /(n-1))a_{2}}/n.
Weierstrass' Inequality
If x_{1},x_{2},...x_{n} are n positive real numbers less than 1 and s_{n} = x_{1}+x_{2}+...+x_{n}
Then,
1-s_{n} ≤ (1 - x_{1})(1 - x_{2})...(1 - x_{n})≤ 1/(1+s_{n})
1+s_{n} ≤ (1 + x_{1})(1 + x_{2})...(1 + x_{n})≤ 1/(1-s_{n}) , where it is assumed that s_{n}<1
Chebyshev's Inequality If a_{1},a_{2},...a_{n} , b_{1},b_{2},...b_{n} ∈ R
such that