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A function f(x) satisfies the Lipschitz condition of order beta at x=0 if |f(h)-f(0)|<=B|h|^beta for all |h|<epsilon, where B and beta are independent of h, beta>0, and alpha ...

A function f is said to have a lower bound c if c<=f(x) for all x in its domain. The greatest lower bound is called the infimum.

If x takes only nonnegative values, then P(x>=a)<=(<x>)/a. (1) To prove the theorem, write <x> = int_0^inftyxP(x)dx (2) = int_0^axP(x)dx+int_a^inftyxP(x)dx. (3) Since P(x) is ...

For b>a>0, 1/b<(lnb-lna)/(b-a)<1/a.

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], ...

Let f(z) be an analytic function in an angular domain W:|argz|<alphapi/2. Suppose there is a constant M such that for each epsilon>0, each finite boundary point has a ...

Let f(x) be a nonnegative and monotonic decreasing function in [a,b] and g(x) such that 0<=g(x)<=1 in [a,b], then int_(b-k)^bf(x)dx<=int_a^bf(x)g(x)dx<=int_a^(a+k)f(x)dx, ...

An inequality is strict if replacing any "less than" and "greater than" signs with equal signs never gives a true expression. For example, a<=b is not strict, whereas a<b is.

phi(A)+phi(B)-phi(A union B)>=phi(A intersection B).

A function f is said to have a upper bound C if f(x)<=C for all x in its domain. The least upper bound is called the supremum. A set is said to be bounded from above if it ...