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# Common inequalities

Source:
The Concise Oxford Dictionary of Mathematics

# Common inequalities

 Triangle inequality Means arithmetic mean≥geometric mean≥harmonic mean where x1, x2,…, xn are non‐negative real numbers Abel’s where {fi} is a sequence of real numbers with f i+1≥fi and {ai} a sequence of real or complex numbers, and A = max {|a 1|,|a 2|,…|an|}. Bernoulli’s (1+x)n≥1+nx for integers n>1 and any real number x>−1. Cauchy–Schwarz inequality for integrals If f(x), g(x) are real functions then if all these integrals exist. Cauchy–Schwarz inequality for sums If ai and bi are real numbers, i=1, 2, …, n then Chebyshev’s inequalities If X is a random variable and g(X) is always ≥0 then Holder’s for integrals Holder’s for sums Isoperimetric inequality If p is the perimeter of a closed curve in a plane and the area enclosed by the curve is A, then p 2≤4 πA