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If the fourth moment mu_4!=0, then P(|x^_-mu_4|>=lambda)<=(mu_4+3(N-1)sigma^4)/(N^3lambda^4), where sigma^2 is the variance.

If 0<=g(x)<=1 and g is nonincreasing on the interval [0, 1], then for all possible values of a and b, int_0^1g(x^(1/(a+b)))dx>=int_0^1g(x^(1/a))dxint_0^1g(x^(1/b))dx.

For homogeneous polynomials P and Q of degree m and n, then sqrt((m!n!)/((m+n)!))[P]_2[Q]_2<=[P·Q]_2<=[P]_2[Q]_2, where [P·Q]_2 is the Bombieri norm.

Gives a lower bound for the inner product (Lu,u), where L is a linear elliptic real differential operator of order m, and u has compact support.

Let f_1(x), ..., f_n(x) be real integrable functions over the closed interval [a,b], then the determinant of their integrals satisfies

Let A=a_(ik) be an arbitrary n×n nonsingular matrix with real elements and determinant |A|, then |A|^2<=product_(i=1)^n(sum_(k=1)^na_(ik)^2).

Let V be an inner product space and let x,y,z in V. Hlawka's inequality states that ||x+y||+||y+z||+||z+x||<=||x||+||y||+||z||+||x+y+z||, where the norm ||z|| denotes the ...

Let a plane figure have area A and perimeter p. Then Q=(4piA)/(p^2)<=1, where Q is known as the isoperimetric quotient. The equation becomes an equality only for a circle.

Given a convex plane region with area A and perimeter p, then |N-A|<p, where N is the number of enclosed lattice points.

If f(x) is a monotonically increasing integrable function on [a,b] with f(b)<=0, then if g is a real function integrable on [a,b], ...