Minkowski Inequality - Minkowski's Integral Inequality

Minkowski's Integral Inequality

Suppose that (S1,μ1) and (S2,μ2) are two measure spaces and F : S1×S2 → R is measurable. Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood & Pólya 1988, Theorem 202):

with obvious modifications in the case p = ∞. If p > 1, and both sides are finite, then equality holds only if |F(x,y)| = φ(x)ψ(y) a.e. for some non-negative measurable functions φ and ψ.

If μ1 is the counting measure on a two-point set S1 = {1,2}, then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting ƒi(y) = F(i,y) for i = 1,2, the integral inequality gives

\begin{align} \|f_1 + f_2\|_p &= \left^{1/p} \le\int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,d\mu_2(y)\right)^{1/p}d\mu_1(x)=\|f_1\|_p + \|f_2\|_p. \end{align}

Read more about this topic:  Minkowski Inequality

Famous quotes containing the words integral and/or inequality:

    “Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an integral piece—is being torn from him.”
    —Lawrence Balter (20th century)

    “However energetically society in general may strive to make all the citizens equal and alike, the personal pride of each individual will always make him try to escape from the common level, and he will form some inequality somewhere to his own profit.”
    —Alexis de Tocqueville (1805–1859)