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Common inequalities

The Concise Oxford Dictionary of Mathematics

Common inequalities

Triangle inequality

Common inequalities


arithmetic mean≥geometric mean≥harmonic mean Common inequalities where x1, x2,…, xn are non‐negative real numbers


Common inequalities where {fi} is a sequence of real numbers with f i+1fi and {ai} a sequence of real or complex numbers, and A = max {|a 1|,|a 2|,…|an|}.


(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 Common inequalities Common inequalities if all these integrals exist.

Cauchy–Schwarz inequality for sums

If ai and bi are real numbers, i=1, 2, …, n then Common inequalities


Common inequalities


If X is a random variable and g(X) is always ≥0 then Common inequalities

Holder’s for integrals

Common inequalities

Holder’s for sums

Common inequalities

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