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If a_1>=a_2>=...>=a_n (1) b_1>=b_2>=...>=b_n, (2) then nsum_(k=1)^na_kb_k>=(sum_(k=1)^na_k)(sum_(k=1)^nb_k). (3) This is true for any distribution.

A cylindrical algebraic decomposition that omits sets of measure zero. Generic cylindrical algebraic decompositions are generally much quicker to compute than are normal ...

A quantity a is said to be greater than b if a is larger than b, written a>b. If a is greater than or equal to b, the relationship is written a>=b. In the Wolfram Language, ...

The identric mean is defined by I(a,b)=1/e((b^b)/(a^a))^(1/(b-a)) for a>0, b>0, and a!=b. The identric mean has been investigated intensively and many remarkable inequalities ...

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.

A quantity a is said to be less than b if a is smaller than b, written a<b. If a is less than or equal to b, the relationship is written a<=b. In the Wolfram Language, this ...

A necessary and sufficient condition that [alpha^'] should be comparable with [alpha] for all positive values of the a is that one of (alpha^') and (alpha) should be ...

Let {f_n} and {a_n} be sequences with f_n>=f_(n+1)>0 for n=1, 2, ..., then |sum_(n=1)^ma_nf_n|<=Af_1, where A=max{|a_1|,|a_1+a_2|,...,|a_1+a_2+...+a_m|}.

If f is continuous on a closed interval [a,b], then there is at least one number x^* in [a,b] such that int_a^bf(x)dx=f(x^*)(b-a). The average value of the function (f^_) on ...

For c<1, x^c<1+c(x-1). For c>1, x^c>1+c(x-1).